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Abstract:

Apparatus, methods, and systems provide conversion of evanescent
electromagnetic waves to non-evanescent electromagnetic waves and/or
conversion of non-evanescent electromagnetic waves to evanescent
electromagnetic waves. In some approaches the conversion includes
propagation of electromagnetic waves within an indefinite electromagnetic
medium, and the indefinite medium may include an artificially-structured
material such as a layered structure or other metamaterial.

Claims:

1. An electromagnetic apparatus, comprising:a conversion structure with
electromagnetic parameters that include indefinite tensor parameters, the
conversion structure being responsive to an evanescent electromagnetic
wave at a first surface region to provide a non-evanescent
electromagnetic wave at a second surface region, where the first and
second surface regions are substantially nonplanar and substantially
non-concentric.

2. The electromagnetic apparatus of claim 1, wherein the first surface
region has a non-constant curvature.

3. The electromagnetic apparatus of claim 2, wherein the first surface
region includes a substantially convex surface subregion and a
substantially concave surface subregion.

4. The electromagnetic apparatus of claim 1, wherein the second surface
region has a non-constant curvature.

5. The electromagnetic apparatus of claim 5, wherein the second surface
region includes a substantially convex surface subregion and a
substantially concave surface subregion.

6. The electromagnetic apparatus of claim 1, wherein the first and second
surface regions are substantially concave.

7. The electromagnetic apparatus of claim 1, wherein the first and second
surface regions are substantially convex.

8. The electromagnetic apparatus of claim 1, wherein a center of a first
osculating circle of the first surface region is different than a center
of a second osculating circle of the second surface region, the second
osculating circle being coplanar with the first osculating circle.

9. The electromagnetic apparatus of claim 8, wherein the first surface
region is substantially concave and the second surface region is
substantially convex.

10. The electromagnetic apparatus of claim 8, wherein the first surface
region is substantially convex and the second surface region is
substantially concave.

20. The electromagnetic apparatus of claim 1, wherein the conversion
structure is responsive to the evanescent electromagnetic wave at a first
frequency with a first wavenumber to provide the non-evanescent
electromagnetic wave at the first frequency with a second wavenumber, the
first wavenumber corresponding to a surface parallel direction of the
first surface region and the second wavenumber corresponding to a surface
parallel direction of the second surface region.

21. The electromagnetic apparatus of claim 20, wherein a first region
outside the conversion structure and adjacent to the first surface region
defines a first phase velocity for electromagnetic radiation, and the
first wavenumber is greater than the first frequency divided by the first
phase velocity.

22. (canceled)

23. The electromagnetic apparatus of claim 21, wherein a second region
outside the conversion structure and adjacent to the second surface
region defines a second phase velocity for electromagnetic radiation, and
the second wavenumber is less than the first frequency divided by the
second phase velocity.

24-28. (canceled)

29. The electromagnetic apparatus of claim 1, wherein a separation
distance between the first and second surface regions is substantially
greater than an evanescent range of the evanescent electromagnetic wave.

30. The electromagnetic apparatus of claim 1, wherein the conversion
structure includes a succession of adjacent layers.

31. The electromagnetic apparatus of claim 30, where a first layer in the
succession of adjacent layers substantially coincides with the first
surface region

32. The electromagnetic apparatus of claim 30, where a last layer in the
succession of adjacent layers substantially coincides with the second
surface region.

33. (canceled)

34. The electromagnetic apparatus of claim 30, wherein the succession of
adjacent layers is an alternating succession of adjacent layers having
first and second alternate electromagnetic properties.

35. The electromagnetic apparatus of claim 34, where the first alternate
electromagnetic properties include a first permittivity greater than
zero, and the second alternate electromagnetic properties include a
second permittivity less than zero.

36. The electromagnetic apparatus of claim 34, where the first alternate
electromagnetic properties include a first permeability greater than
zero, and the second alternate electromagnetic properties include a
second permeability less than zero.

37. (canceled)

38. The electromagnetic apparatus of claim 30, wherein the succession of
adjacent layers is an alternating succession of adjacent layers of first
and second alternate materials.

39. The electromagnetic apparatus of claim 38, wherein the first alternate
material is a metal and the second alternate material is a dielectric.

40-42. (canceled)

43. The electromagnetic apparatus of claim 38, wherein the first alternate
material is a metal and the second alternate material is a gain medium.

44. (canceled)

45. (canceled)

46. The electromagnetic apparatus of claim 38, wherein the first alternate
material is a first semiconductor material and the second alternative
material is a second semiconductor material.

51. The electromagnetic apparatus of claim 49, wherein the axial direction
is a non-constant axial direction that is a function of location within
the conversion structure.

52. The electromagnetic apparatus of claim 49, wherein the conversion
structure is responsive to the evanescent electromagnetic wave at the
first surface region to convey a propagating electromagnetic wave from
the first surface region to the second surface region along a propagation
direction that substantially coincides with the axial direction.

53. The electromagnetic apparatus of claim 49, wherein the conversion
structure is responsive to the evanescent electromagnetic wave at the
first surface region to convey a propagating electromagnetic wave from
the first surface region to the second surface region along at least two
propagation directions, each of the at least two propagation directions
substantially having a common angle with respect to the axial direction.

54-56. (canceled)

57. The electromagnetic apparatus of claim 49, wherein the axial
electromagnetic parameters include an axial permittivity, and the
transverse electromagnetic parameters include a transverse permeability
and a transverse permittivity.

58. The electromagnetic apparatus of claim 57, wherein the axial
permittivity is less than zero, the transverse permittivity is greater
than or equal to zero, and the transverse permeability is greater than or
equal to zero.

59. The electromagnetic apparatus of claim 58, wherein the transverse
permittivity is substantially equal to zero.

60. The electromagnetic apparatus of claim 58, wherein the axial
permittivity is substantially less than minus one.

61. The electromagnetic apparatus of claim 58, wherein the transverse
permeability is substantially equal to one.

62. The electromagnetic apparatus of claim 57, wherein the axial
permittivity is greater than zero, the transverse permittivity is less
than or equal to zero, and the transverse permeability is less than or
equal to zero.

63. The electromagnetic apparatus of claim 62, wherein the transverse
permittivity is substantially equal to zero.

64. The electromagnetic apparatus of claim 62, wherein the axial
permittivity is substantially greater than one.

65. The electromagnetic apparatus of claim 49, wherein the axial
electromagnetic parameters include an axial permeability, and the
transverse electromagnetic parameters include a transverse permeability
and a transverse permittivity.

66. The electromagnetic apparatus of claim 65, wherein the axial
permeability is less than zero, the transverse permittivity is greater
than or equal to zero, and the transverse permeability is greater than or
equal to zero.

67. The electromagnetic apparatus of claim 66, wherein the transverse
permeability is substantially equal to zero.

68. The electromagnetic apparatus of claim 66, wherein the axial
permeability is substantially less than minus one.

69. The electromagnetic apparatus of claim 65, wherein the axial
permeability is greater than zero, the transverse permittivity is less
than or equal to zero, and the transverse permeability is less than or
equal to zero.

70. The electromagnetic apparatus of claim 69, wherein the transverse
permeability is substantially equal to zero.

71. The electromagnetic apparatus of claim 69, wherein the axial
permeability is substantially greater than one.

72. The electromagnetic apparatus of claim 57, wherein the conversion
structure includes an artificially-structured material having an
effective electromagnetic response that corresponds to the
electromagnetic parameters.

73-76. (canceled)

77. The electromagnetic apparatus of claim 72, wherein the
artificially-structured material includes a metamaterial.

78-85. (canceled)

86. The electromagnetic apparatus of claim 77, wherein the metamaterial
includes a plurality of electromagnetically responsive elements disposed
at a plurality of spatial locations and having a plurality of individual
responses, the plurality of individual responses composing the effective
electromagnetic response.

272. An electromagnetic apparatus, comprising:a conversion structure with
electromagnetic parameters that include indefinite tensor parameters, the
conversion structure being responsive to a non-evanescent electromagnetic
wave at a first surface region to provide an evanescent electromagnetic
wave at a second surface region, where the first and second surface
regions are substantially nonplanar and substantially non-concentric.

Description:

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001]The present application is related to and claims the benefit of the
earliest available effective filing date(s) from the following listed
application(s) (the "Related Applications") (e.g., claims earliest
available priority dates for other than provisional patent applications
or claims benefits under 35 USC §119(e) for provisional patent
applications, for any and all parent, grandparent, great-grandparent,
etc. applications of the Related Application(s)). All subject matter of
the Related Applications and of any and all parent, grandparent,
great-grandparent, etc. applications of the Related Applications is
incorporated herein by reference to the extent such subject matter is not
inconsistent herewith.

[0004]The United States Patent Office (USPTO) has published a notice to
the effect that the USPTO's computer programs require that patent
applicants reference both a serial number and indicate whether an
application is a continuation or continuation-in-part. Stephen G. Kunin,
Benefit of Prior-Filed Application, USPTO Official Gazette Mar. 18, 2003,
available at
http://www.uspto.gov/web/offices/com/sol/og/2003/week11/patbene.htm. The
present Applicant Entity (hereinafter "Applicant") has provided above a
specific reference to the application(s)from which priority is being
claimed as recited by statute. Applicant understands that the statute is
unambiguous in its specific reference language and does not require
either a serial number or any characterization, such as "continuation" or
"continuation-in-part," for claiming priority to U.S. patent
applications. Notwithstanding the foregoing, Applicant understands that
the USPTO's computer programs have certain data entry requirements, and
hence Applicant is designating the present application as a
continuation-in-part of its parent applications as set forth above, but
expressly points out that such designations are not to be construed in
any way as any type of commentary and/or admission as to whether or not
the present application contains any new matter in addition to the matter
of its parent application(s).

[0006]FIG. 1 depicts a conversion structure having first and second
surface regions that are substantially planar and substantially parallel.

[0007]FIG. 2 depicts a layered structure as an exemplary implementation of
the conversion structure of FIG. 1.

[0008]FIG. 3 depicts a layered structure as an exemplary implementation of
the conversion structure of FIG. 1.

[0009]FIG. 4 depicts a conversion structure having a first surface region
that is substantially planar and a second surface region that is
substantially nonplanar.

[0010]FIG. 5 depicts a layered structure as an exemplary implementation of
the conversion structure of FIG. 4.

[0011]FIG. 6 depicts a layered structure as an exemplary implementation of
the conversion structure of FIG. 4.

[0012]FIG. 7 depicts a conversion structure having a first surface region
that is substantially nonplanar and a second surface region that is
substantially planar.

[0013]FIG. 8 depicts a layered structure as an exemplary implementation of
the conversion structure of FIG. 7.

[0014]FIG. 9 depicts a layered structure as an exemplary implementation of
the conversion structure of FIG. 7.

[0015]FIG. 10 depicts various conversion structures having first and
second surface regions that are substantially nonplanar.

[0016]FIG. 11 depicts a layered structure as an exemplary implementation
of a conversion structure as in FIG. 10.

[0017]FIG. 12 depicts a first process flow.

[0018]FIG. 13 depicts a second process flow reciprocal to the first
process flow.

[0019]FIG. 14 depicts a system that includes an evanescent conversion
unit.

DETAILED DESCRIPTION

[0020]In the following detailed description, reference is made to the
accompanying drawings, which form a part hereof. In the drawings, similar
symbols typically identify similar components, unless context dictates
otherwise. The illustrative embodiments described in the detailed
description, drawings, and claims are not meant to be limiting. Other
embodiments may be utilized, and other changes may be made, without
departing from the spirit or scope of the subject matter presented here.

[0021]Embodiments provide apparatus and methods for converting evanescent
electromagnetic waves to non-evanescent electromagnetic waves and/or for
converting non-evanescent electromagnetic waves to evanescent
electromagnetic waves. In general, an evanescent electromagnetic wave is
an electromagnetic wave having an amplitude that decays exponentially
with distance, e.g. having a wave vector that is at least partially
imaginary. For example, the electric component of an electromagnetic wave
may have a 2D Fourier expansion given by

Supposing, for purposes of illustration, that the wave exists in a medium
with refractive index n, the Fourier modes having
kx2+ky2<n2ω2/c2 are
propagating electromagnetic waves with real wavevector components
kz=+ {square root over
(n2ω2c-2-kx2-ky2)}, while the
Fourier modes having
kx2+ky2>n2ω2/c2 are
evanescent electromagnetic waves with imaginary wavevector components
kz=+i {square root over
(kx2+ky2-n2ω2c-2)}. The
evanescent electromagnetic waves decay exponentially with distance z. In
a conventional far-field optics application, where z may represent, for
example, distance from an object plane of a conventional far-field
optical system, the evanescent waves do not substantially persist beyond
an evanescent range μ˜1/|kz|, corresponding to a near field
of the object plane (or a near field in the vicinity of an object to be
imaged), while the propagating waves persist beyond the near field into
the far field to comprise a far-field image (e.g. on an image plane of
the conventional far-field optical system). Thus, a conventional
far-field optical system has a resolution limit Δ (sometimes
referred to as a diffraction limit or an Abbe-Rayleigh limit)
corresponding to a maximum transverse wavevector kmax for
propagating waves:

Δ ~ 2 π k max = 2 π c n
ω = c nv = λ 0 n ( 2 ) ##EQU00002##

where λ0 is the free-space wavelength corresponding to
frequency v. On the other hand, embodiments disclosed herein provide
apparatus and methods that may exceed this resolution limit, by
converting evanescent waves to propagating waves (or vice versa) in an
indefinite electromagnetic medium.

[0022]In general, an indefinite electromagnetic medium is an
electromagnetic medium having electromagnetic parameters (e.g.
permittivity and/or permeability) that include indefinite tensor
parameters. Throughout this disclosure, including the subsequent claims,
the term "indefinite" is taken to have its algebraic meaning; thus, an
indefinite tensor is a tensor that is neither positive definite (having
all positive eigenvalues) nor negative definite (having all negative
eigenvalues) but instead has at least one positive eigenvalue and at
least one negative eigenvalue. Some exemplary indefinite media are
described in D. R. Smith and D. Schurig, "Indefinite materials," U.S.
patent application Ser. No. 10/525,191; D. R. Smith and D. Schurig,
"Electromagnetic wave propagation in media with indefinite permittivity
and permeability tensors," Phys. Rev. Lett. 90, 077405 (2003); and D. R.
Smith and D. Schurig, "Sub-diffraction imaging with compensating
bilayers," New. J. Phys. 7, 162 (2005); each of which is herein
incorporated by reference.

[0023]In some embodiments an indefinite medium is an electromagnetic
medium having an indefinite permeability. An example of an indefinite
permeability medium is a planar slab having a z-axis perpendicular to the
slab (with x- and y-axes parallel to the slab) and electromagnetic
parameters εy, μx, and μz satisfying the
inequalities

εyμx>0, μx/μz<0 (3)

(thus, the permeability is indefinite, with either
μx<0≦μz or μx>0>μz). For
TE-polarized (i.e. s-polarized) electromagnetic waves with an electric
field directed along the y-axis, these electromagnetic parameters provide
a hyperbolic dispersion relation

k z 2 = y μ x ω 2 c 2 - μ x μ
z k x 2 ( 4 ) ##EQU00003##

that admits propagating electromagnetic waves (real kz) with large
transverse wavevectors kx. Thus, if the planar slab adjoins a
uniform refractive medium with index of refraction n, an evanescent wave
in the adjoining medium (e.g. as in equation (1), with
kx>nω/c) becomes a propagating wave in the indefinite
medium (or, reciprocally, a propagating wave in the indefinite medium
becomes an evanescent wave in the adjoining medium). For sufficiently
large kx (i.e. substantially within the asymptotic domain of the
hyperbolic dispersion relation (4)), the propagating wave is
characterized by group velocities that are substantially perpendicular to
the asymptotes of equation (4), i.e. the propagating wave is
substantially conveyed along propagation directions in the xz-plane that
form an angle θx=tan-1(|μx/μz|) with
respect to the z-axis (e.g. as depicted in FIG. 10 of the previously
cited U.S. patent application Ser. No. 10/525,191); moreover, for
sufficiently small μx (i.e. |μx| substantially equal to
zero and/or substantially less than |μz|), the angle
θx becomes substantially equal to zero and the multiple
propagating directions degenerate to a single propagation direction that
substantially coincides with the z-axis (in this case the indefinite
medium shall be referred to as a degenerate indefinite medium). The
planar slab may alternately or additionally have electromagnetic
parameters εx and μy, satisfying the alternate or
additional inequalities

εxμy>0, μy/μz<0, (5)

providing another hyperbolic dispersion relation

k z 2 = x μ y ω 2 c 2 - μ y μ
z k y 2 ( 6 ) ##EQU00004##

for TE-polarized electromagnetic waves with an electric field directed
along the x-axis. In this case, for sufficiently large ky (i.e.
substantially within the asymptotic domain of the hyperbolic dispersion
relation (6)), a propagating wave in the indefinite medium is
characterized by group velocities that are substantially perpendicular to
the asymptotes of equation (6), i.e. the propagating wave is
substantially conveyed along propagation directions in the yz-plane that
form an angle θy=tan-1(|μy/μz|) with
respect to the z-axis; moreover, for sufficiently small μy (i.e.
|μy| substantially equal to zero and/or substantially less than
|μz|), the angle θy becomes substantially equal to
zero and the multiple propagating directions degenerate to a single
propagation direction that substantially coincides with the z-axis
(another degenerate indefinite medium). When the planar slab satisfies
both inequalities (3) and (5), the indefinite medium supports
TE-polarized waves that substantially propagate (for sufficiently large
transverse wavevectors kx and/or ky) along propagation
directions that compose an elliptical cone having a cone axis that
coincides with the z-direction and half-angles θx and
θy, as above, with respect to the x- and y-axes, and in the
case where εx=εy and μx=μy, the
planar slab is a uniaxial medium that provides the same hyperbolic
dispersion for any TE-polarized waves, and the propagation directions for
large transverse wavevectors compose a circular cone with
θx=θy.

[0024]More generally, in some embodiments an indefinite permeability
medium may define an axial direction that corresponds to a first
eigenvector of the indefinite permeability matrix, with first and second
transverse directions that correspond to second and third eigenvectors of
the indefinite permeability matrix, respectively. The parameters of the
indefinite permeability matrix may vary with position within the
indefinite permeability medium, and correspondingly the eigenvectors of
the indefinite permeability matrix may also vary with position within the
indefinite permeability medium. The disclosure of the preceding paragraph
may encompass more general embodiments of an indefinite permeability
medium, in the following manner: the z-axis shall be understood to refer
more generally to an axial direction that may vary throughout the
indefinite medium, the x-axis shall be understood to refer more generally
to a first transverse direction perpendicular to the axial direction, and
the y-axis shall be understood to refer more generally to a second
transverse direction mutually perpendicular to the axial direction and
the first transverse direction. Thus, for example, a uniaxial indefinite
permeability medium may have a local axial parameter μA
(corresponding to an axial direction that may vary with position within
the medium) and transverse parameters
εT1=εT2=εT,
μT1=μT2=μT that satisfy the inequalities

εTμT>0, μT/μA<0, (7)

providing a hyperbolic dispersion relation

k A 2 = T μ T ω 2 c 2 - μ T μ
A k T 2 , ( 8 ) ##EQU00005##

and this dispersion relation supports TE-polarized waves that
substantially propagate (for sufficiently large transverse wavevectors
kT) along propagation directions that locally compose a circular
cone having a cone axis that coincides with the local axial direction
with a cone half-angle θ=tan-1(|μT/μA|) (and
where |μT|<<|μA|, the medium is a degenerate
indefinite medium, wherein the cone of propagation directions degenerates
to a single propagation direction that substantially coincides with the
local axial direction).

[0025]In some embodiments an indefinite medium is an electromagnetic
medium having an indefinite permittivity. An example of an indefinite
permittivity medium is a planar slab having a z-axis perpendicular to the
slab (with x- and y-axes parallel to the slab), and having
electromagnetic parameters μy, εy, and
εz satisfying the inequalities

μyεx>0, εx/εz<0 (9)

(thus, the permittivity is indefinite, with either
εx<0<εz or
εx>0>εz). For TM-polarized (i.e.
p-polarized) electromagnetic waves with a magnetic field directed along
the y-axis, these electromagnetic parameters provide a hyperbolic
dispersion relation

k z 2 = μ y x ω 2 c 2 - x z k
x 2 ( 10 ) ##EQU00006##

that admits propagating electromagnetic waves (real kz) with large
transverse wavevectors kz. Thus, if the planar slab adjoins a
uniform refractive medium with index of refraction n, an evanescent wave
in the adjoining medium (e.g. as in equation (1), with
kx>nω/c) becomes a propagating wave in the indefinite
medium (or, reciprocally, a propagating wave in the indefinite medium
becomes an evanescent wave in the adjoining medium). For sufficiently
large kz (i.e. substantially within the asymptotic domain of the
hyperbolic dispersion relation (10)), the propagating wave is
characterized by group velocities that are substantially perpendicular to
the asymptotes of equation (10), i.e. the propagating wave is
substantially conveyed along propagation directions in the xz-plane that
form an angle θx=tan-1(|εx/εz|)
with respect to the z-axis; moreover, for sufficiently small
εx (i.e. |εx| substantially equal to zero
and/or substantially less than |εz|), the angle
θx becomes substantially equal to zero and the multiple
propagating directions degenerate to a single propagation direction that
substantially coincides with the z-axis (in this case the indefinite
medium shall be referred to as a degenerate indefinite medium). The
planar slab may alternately or additionally have electromagnetic
parameters μx and εy, satisfying the alternative or
additional inequalities

μxεy>0, εy/εz<0,
(11)

providing another hyperbolic dispersion relation

k z 2 = μ x y ω 2 c 2 - y z k
y 2 ( 12 ) ##EQU00007##

for TM-polarized electromagnetic waves with a magnetic field directed
along the x-axis. In this case, for sufficiently large ky (i.e.
substantially within the asymptotic domain of the hyperbolic dispersion
relation (12)), a propagating wave in the indefinite medium is
characterized by group velocities that are substantially perpendicular to
the asymptotes of equation (12), i.e. the propagating wave is
substantially conveyed along propagation directions in the yz-plane that
form an angle θy=tan-1(|εy/εz|)
with respect to the z-axis; moreover, for sufficiently small
εy (i.e. |εy| substantially equal to zero
and/or substantially less than |εz|), the angle
θy becomes substantially equal to zero and the multiple
propagation directions degenerate to a single propagation direction that
substantially coincides with the z-axis (another degenerate indefinite
medium). When the planar slab satisfies both inequalities (9) and (11),
the indefinite medium supports TM-polarized waves that substantially
propagate (for sufficiently large transverse wavevectors kx and/or
ky) along propagation directions that compose an elliptical cone
having a cone axis that coincides with the z-direction and half-angles
θx and θy, as above, with respect to the x- and
y-axes, and in the case where εx=εy and
μx=μy, the planar slab is a uniaxial medium that provides
the same hyperbolic dispersion for any TM-polarized waves, and the
propagation directions for large transverse wavevectors compose a
circular cone with θx=θy.

[0026]More generally, in some embodiments an indefinite permittivity
medium may define an axial direction that corresponds to a first
eigenvector of the indefinite permittivity matrix, with first and second
transverse directions that correspond to second and third eigenvectors of
the indefinite permittivity matrix, respectively. The parameters of the
indefinite permittivity matrix may vary with position within the
indefinite permittivity medium, and correspondingly the eigenvectors of
the indefinite permittivity matrix may also vary with position within the
indefinite permittivity medium. The disclosure of the preceding paragraph
may encompass more general embodiments of an indefinite permittivity
medium, in the following manner: the z-axis shall be understood to refer
more generally to an axial direction that may vary throughout the
indefinite medium, the x-axis shall be understood to refer more generally
to a first transverse direction perpendicular to the axial direction, and
the y-axis shall be understood to refer more generally to a second
transverse direction mutually perpendicular to the axial direction and
the first transverse direction. Thus, for example, a uniaxial indefinite
permittivity medium may have a local axial parameter εA
(corresponding to an axial direction that may vary with position within
the medium) and transverse parameters
εT1=εT2=εT,
μT1=μT2=μT that satisfy the inequalities

εTμT>0, εT/εA<0,
(13)

providing a hyperbolic dispersion relation

k A 2 = T μ T ω 2 c 2 - T A
k T 2 , ( 14 ) ##EQU00008##

and this dispersion relation supports TM-polarized waves that
substantially propagate (for sufficiently large transverse wavevectors
kT) along propagation directions that locally compose a circular
cone having a cone axis that coincides with the local axial direction
with a cone half-angle
θ=tan-1(|εT/εA|) (and where
|εT|<<|εA|, the medium is a degenerate
indefinite medium, wherein the cone of propagation directions degenerates
to a single propagation direction that substantially coincides with the
local axial direction).

[0027]In some embodiments an indefinite medium is an electromagnetic
medium that is "doubly indefinite," i.e. having both an indefinite
permittivity and an indefinite permeability. An example of a doubly
indefinite medium is a planar slab defining a z-axis perpendicular to the
slab (with x- and y-axes parallel to the slab), and having
electromagnetic parameters satisfying one or both of equations (3) and
(5) (providing indefinite permeability) and one or both of equations (9)
and (11) (providing indefinite permittivity). The doubly-indefinite
planar slab provides a hyperbolic dispersion relation for at least one
TE-polarized wave (as in equations (4) and/or (6)) and further provides a
hyperbolic dispersion relation for at least one TM-polarized wave (as in
equations (10) and (12)), with wave propagation features as discussed in
the preceding paragraphs containing the equations that are referenced
here.

[0028]In some embodiments a doubly-indefinite medium may have an
indefinite permittivity matrix and an indefinite permeability matrix that
are substantially simultaneously diagonalizable, and the
doubly-indefinite medium defines an axial direction that corresponds to a
first common eigenvector of the indefinite matrices, with first and
second transverse directions that correspond to second and third common
eigenvectors of the indefinite matrices, respectively. As in the
preceding examples, the parameters of the indefinite matrices may vary
with position within the doubly-indefinite medium, and correspondingly
the common eigenvectors of the indefinite matrices may also vary with
position within the doubly-indefinite medium. The disclosure of the
preceding paragraph may encompass more general embodiments of a
doubly-indefinite medium, in the following manner: the z-axis shall be
understood to refer more generally to an axial direction that may vary
throughout the doubly-indefinite medium, the x-axis shall be understood
to refer more generally to a first transverse direction perpendicular to
the axial direction, and the y-axis shall be understood to refer more
generally to a second transverse direction mutually perpendicular to the
axial direction and the first transverse direction. Thus, for example, a
uniaxial doubly-indefinite medium may have local axial parameters
εA, μA (corresponding to an axial direction that may
vary with position within the medium) and transverse parameters
εT1=εT2=εT,
μT1=μT2=μT that satisfy the inequalities (7) and
(13), providing hyperbolic dispersion relations (8) and (14), and these
dispersion relations respectively support TE- and TM-polarized waves
within the doubly-indefinite medium, as discussed in the preceding
paragraphs containing the equations that are referenced here.

[0029]Some embodiments provide an indefinite medium that is a
transformation medium, i.e. an electromagnetic medium having properties
that may be characterized according to transformation optics.
Transformation optics is an emerging field of electromagnetic
engineering, and transformation optics devices include structures that
influence electromagnetic waves, where the influencing imitates the
bending of electromagnetic waves in a curved coordinate space (a
"transformation" of a flat coordinate space), e.g. as described in A. J.
Ward and J. B. Pendry, "Refraction and geometry in Maxwell's equations,"
J. Mod. Optics 43, 773 (1996), J. B. Pendry and S. A. Ramakrishna,
"Focusing light using negative refraction," J. Phys. [Cond. Matt.] 15,
6345 (2003), D. Schurig et al, "Calculation of material properties and
ray tracing in transformation media," Optics Express 14, 9794 (2006) ("D.
Schurig et al (1)"), and in U. Leonhardt and T. G. Philbin, "General
relativity in electrical engineering," New J. Phys. 8, 247 (2006), each
of which is herein incorporated by reference. The use of the term
"optics" does not imply any limitation with regards to wavelength; a
transformation optics device may be operable in wavelength bands that
range from radio wavelengths to visible wavelengths and beyond.

[0030]A first exemplary transformation optics device is the
electromagnetic cloak that was described, simulated, and implemented,
respectively, in J. B. Pendry et al, "Controlling electromagnetic waves,"
Science 312, 1780 (2006); S. A. Cummer et al, "Full-wave simulations of
electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006); and
D. Schurig et al, "Metamaterial electromagnetic cloak at microwave
frequencies," Science 314, 977 (2006) ("D. Schurig et al (2)"); each of
which is herein incorporated by reference. See also J. Pendry et al,
"Electromagnetic cloaking method," U.S. patent application Ser. No.
11/459,728, herein incorporated by reference. For the electromagnetic
cloak, the curved coordinate space is a transformation of a flat space
that has been punctured and stretched to create a hole (the cloaked
region), and this transformation corresponds to a set of constitutive
parameters (electric permittivity and magnetic permeability) for a
transformation medium wherein electromagnetic waves are refracted around
the hole in imitation of the curved coordinate space.

[0031]A second exemplary transformation optics device is illustrated by
embodiments of the electromagnetic compression structure described in J.
B. Pendry, D. Schurig, and D. R. Smith, "Electromagnetic compression
apparatus, methods, and systems," U.S. patent application Ser. No.
11/982,353; and in J. B. Pendry, D. Schurig, and D. R. Smith,
"Electromagnetic compression apparatus, methods, and systems," U.S.
patent application Ser. No. 12/069,170; each of which is herein
incorporated by reference. In embodiments described therein, an
electromagnetic compression structure includes a transformation medium
with constitutive parameters corresponding to a coordinate transformation
that compresses a region of space intermediate first and second spatial
locations, the effective spatial compression being applied along an axis
joining the first and second spatial locations. The electromagnetic
compression structure thereby provides an effective electromagnetic
distance between the first and second spatial locations greater than a
physical distance between the first and second spatial locations.

[0032]A third exemplary transformation optics device is illustrated by
embodiments of the electromagnetic cloaking and/or translation structure
described in J. T. Kare, "Electromagnetic cloaking apparatus, methods,
and systems," U.S. patent application Ser. No. 12/074,247; and in J. T.
Kare, "Electromagnetic cloaking apparatus, methods, and systems," U.S.
patent application Ser. No. 12/074,248; each of which is herein
incorporated by reference. In embodiments described therein, an
electromagnetic translation structure includes a transformation medium
that provides an apparent location of an electromagnetic transducer
different then an actual location of the electromagnetic transducer,
where the transformation medium has constitutive parameters corresponding
to a coordinate transformation that maps the actual location to the
apparent location. Alternatively or additionally, embodiments include an
electromagnetic cloaking structure operable to divert electromagnetic
radiation around an obstruction in a field of regard of the transducer
(and the obstruction can be another transducer).

[0035]In general, for a selected coordinate transformation, a
transformation medium can be identified wherein electromagnetic fields
evolve as in a curved coordinate space corresponding to the selected
coordinate transformation. Constitutive parameters of the transformation
medium can be obtained from the equations:

{tilde over
(ε)}i'j'=[det(Λ)]-1Λii'Λ-
jj'εij (15)

{tilde over
(ε)}i'j'=[det(Λ)]-1Λii'Λ-
jj'μil (16)

where {tilde over (ε)} and {tilde over (μ)} are the
permittivity and permeability tensors of the transformation medium,
ε and μ are the permittivity and permeability tensors of the
original medium in the untransformed coordinate space, and

Λ i i ' = ∂ x i ' ∂ x i
( 17 ) ##EQU00009##

is the Jacobian matrix corresponding to the coordinate transformation. In
some applications, the coordinate transformation is a one-to-one mapping
of locations in the untransformed coordinate space to locations in the
transformed coordinate space, and in other applications the coordinate
transformation is a one-to-many mapping of locations in the untransformed
coordinate space to locations in the transformed coordinate space. Some
coordinate transformations, such as one-to-many mappings, may correspond
to a transformation medium having a negative index of refraction. In some
applications, the transformation medium is an indefinite medium, i.e. an
electromagnetic medium having an indefinite permittivity and/or an
indefinite permeability (these transformation media may be referred to as
"indefinite transformation media"). For example, in equations (15) and
(16), if the original permittivity matrix ε is indefinite, then
the transformed permittivity matrix {tilde over (ε)} is also
indefinite; and/or if the original permeability matrix μ is
indefinite, then the transformed permeability matrix {tilde over (μ)}
is also indefinite. In some applications, only selected matrix elements
of the permittivity and permeability tensors need satisfy equations (15)
and (16), e.g. where the transformation optics response is for a selected
polarization only. In other applications, a first set of permittivity and
permeability matrix elements satisfy equations (15) and (16) with a first
Jacobian Λ, corresponding to a first transformation optics
response for a first polarization of electromagnetic waves, and a second
set of permittivity and permeability matrix elements, orthogonal (or
otherwise complementary) to the first set of matrix elements, satisfy
equations (15) and (16) with a second Jacobian Λ', corresponding
to a second transformation optics response for a second polarization of
electromagnetic waves. In yet other applications, reduced parameters are
used that may not satisfy equations (15) and (16), but preserve products
of selected elements in (15) and selected elements in (16), thus
preserving dispersion relations inside the transformation medium (see,
for example, D. Schurig et al (2), supra, and W. Cai et al, "Optical
cloaking with metamaterials," Nature Photonics 1, 224 (2007), herein
incorporated by reference). Reduced parameters can be used, for example,
to substitute a magnetic response for an electric response, or vice
versa. While reduced parameters preserve dispersion relations inside the
transformation medium (so that the ray or wave trajectories inside the
transformation medium are unchanged from those of equations (15) and
(16)), they may not preserve impedance characteristics of the
transformation medium, so that rays or waves incident upon a boundary or
interface of the transformation medium may sustain reflections (whereas
in general a transformation medium according to equations (15) and (16)
is substantially nonreflective or sustains the reflection characteristics
of the original medium in the untransformed coordinate space). The
reflective or scattering characteristics of a transformation medium with
reduced parameters can be substantially reduced or eliminated (modulo any
reflection characteristics of the original medium in the untransformed
coordinate space) by a suitable choice of coordinate transformation, e.g.
by selecting a coordinate transformation for which the corresponding
Jacobian Λ (or a subset of elements thereof) is continuous or
substantially continuous at a boundary or interface of the transformation
medium (see, for example, W. Cai et al, "Nonmagnetic cloak with minimized
scattering," Appl. Phys. Lett. 91, 111105 (2007), herein incorporated by
reference).

[0036]Embodiments of an indefinite medium and/or a transformation medium
(including embodiments of indefinite transformation media) can be
realized using artificially-structured materials. Generally speaking, the
electromagnetic properties of artificially-structured materials derive
from their structural configurations, rather than or in addition to their
material composition.

[0037]In some embodiments, the artificially-structured materials are
photonic crystals. Some exemplary photonic crystals are described in J.
D. Joannopoulos et al, Photonic Crystals: Molding the Flow of Light,
2nd Edition, Princeton Univ. Press, 2008, herein incorporated by
reference. In a photonic crystals, photonic dispersion relations and/or
photonic band gaps are engineered by imposing a spatially-varying pattern
on an electromagnetic material (e.g. a conducting, magnetic, or
dielectric material) or a combination of electromagnetic materials. The
photonic dispersion relations translate to effective constitutive
parameters (e.g. permittivity, permeability, index of refraction) for the
photonic crystal. The spatially-varying pattern is typically periodic,
quasi-periodic, or colloidal periodic, with a length scale comparable to
an operating wavelength of the photonic crystal.

[0039]Metamaterials generally feature subwavelength elements, i.e.
structural elements with portions having electromagnetic length scales
smaller than an operating wavelength of the metamaterial, and the
subwavelength elements have a collective response to electromagnetic
radiation that corresponds to an effective continuous medium response,
characterized by an effective permittivity, an effective permeability, an
effective magnetoelectric coefficient, or any combination thereof. For
example, the electromagnetic radiation may induce charges and/or currents
in the subwavelength elements, whereby the subwavelength elements acquire
nonzero electric and/or magnetic dipole moments. Where the electric
component of the electromagnetic radiation induces electric
dipole-moments, the metamaterial has an effective permittivity; where the
magnetic component of the electromagnetic radiation induces magnetic
dipole moments, the metamaterial has an effective permeability; and where
the electric (magnetic) component induces magnetic (electric) dipole
moments (as in a chiral metamaterial), the metamaterial has an effective
magnetoelectric coefficient. Some metamaterials provide an artificial
magnetic response; for example, split-ring resonators (SRRs)--or other LC
or plasmonic resonators--built from nonmagnetic conductors can exhibit an
effective magnetic permeability (cf. J. B. Pendry et al, "Magnetism from
conductors and enhanced nonlinear phenomena," IEEE Trans. Micro. Theo.
Tech. 47, 2075 (1999), herein incorporated by reference). Some
metamaterials have "hybrid" electromagnetic properties that emerge
partially from structural characteristics of the metamaterial, and
partially from intrinsic properties of the constituent materials. For
example, G. Dewar, "A thin wire array and magnetic host structure with
n<0," J. Appl. Phys. 97, 10Q101 (2005), herein incorporated by
reference, describes a metamaterial consisting of a wire array
(exhibiting a negative permeability as a consequence of its structure)
embedded in a nonconducting ferrimagnetic host medium (exhibiting an
intrinsic negative permeability). Metamaterials can be designed and
fabricated to exhibit selected permittivities, permeabilities, and/or
magnetoelectric coefficients that depend upon material properties of the
constituent materials as well as shapes, chiralities, configurations,
positions, orientations, and couplings between the subwavelength
elements. The selected permittivites, permeabilities, and/or
magnetoelectric coefficients can be positive or negative, complex (having
loss or gain), anisotropic (including tensor-indefinite), variable in
space (as in a gradient index lens), variable in time (e.g. in response
to an external or feedback signal), variable in frequency (e.g. in the
vicinity of a resonant frequency of the metamaterial), or any combination
thereof. The selected electromagnetic properties can be provided at
wavelengths that range from radio wavelengths to infrared/visible
wavelengths; the latter wavelengths are attainable, e.g., with
nanostructured materials such as nanorod pairs or nano-fishnet structures
(cf. S. Linden et al, "Photonic metamaterials: Magnetism at optical
frequencies," IEEE J. Select. Top. Quant. Elect. 12, 1097 (2006) and V.
Shalaev, "Optical negative-index metamaterials," Nature Photonics 1, 41
(2007), both herein incorporated by reference). An example of a
three-dimensional metamaterial at optical frequencies, an
elongated-split-ring "woodpile" structure, is described in M. S. Rill et
al, "Photonic metamaterials by direct laser writing and silver chemical
vapour deposition," Nature Materials advance online publication, May 11,
2008, (doi:10.1038/nmat2197).

[0040]While many exemplary metamaterials are described as including
discrete elements, some implementations of metamaterials may include
non-discrete elements or structures. For example, a metamaterial may
include elements comprised of sub-elements, where the sub-elements are
discrete structures (such as split-ring resonators, etc.), or the
metamaterial may include elements that are inclusions, exclusions, or
other variations along some continuous structure (e.g. etchings on a
substrate). The metamaterial may include extended structures having
distributed electromagnetic responses (such as distributed inductive
responses, distributed capacitive responses, and distributed
inductive-capacitive responses). Examples include structures consisting
of loaded and/or interconnected transmission lines (such as microstrips
and striplines), artificial ground plane structures (such as artificial
perfect magnetic conductor (PMC) surfaces and electromagnetic band gap
(EGB) surfaces), and interconnected/extended nanostructures
(nano-fishnets, elongated SRR woodpiles, etc.).

[0041]In some embodiments a metamaterial may include a layered structure.
For example, embodiments may provide a structure having a succession of
adjacent layers, where the layers have a corresponding succession of
material properties that include electromagnetic properties (such as
permittivity and/or permeability). The succession of adjacent layers may
be an alternating or repeating succession of adjacent layers, e.g. a
stack of layers of a first type interleaved with layers of a second type,
or a stack that repeats a sequence of three or more types of layers. When
the layers are sufficiently thin (e.g. having thicknesses smaller than an
operating wavelength of the metamaterial), the layered structure may be
characterized as an effective continuous medium having effective
constitutive parameters that relate to the electromagnetic properties of
the individual layers. For example, consider a planar stack of layers of
a first material (of thickness d1, and having homogeneous and
isotropic electromagnetic parameters ε1, μ1)
interleaved with layers of a second material (of thickness d2, and
having homogeneous and isotropic electromagnetic parameters
ε2, μ2); then the layered structure may be
characterized as an effective continuous medium having (effective)
anisotropic constitutive parameters

where η=d2/d1 is the ratio of the two layer thicknesses, z
is the direction normal to the layers, and x, y are the directions
parallel to the layers. When the two materials comprising the interleaved
structure have electromagnetic parameters that are oppositely-signed, the
constitutive parameters (18)-(21) may correspond to an indefinite medium.
For example, when the first material is a material having a permittivity
ε1<0 and the second material is a material having a
permittivity ε2>0, the ratio ηmay be selected to
provide an indefinite permittivity matrix according to equations
(18)-(19) (moreover, for ηsubstantially equal to
|ε1/ε2, the indefinite permittivity medium is
substantially a degenerate indefinite permittivity medium). Alternately
or additionally, when the first material is a material having a
permeability μ1<0 and the second material is a material having
a permeability μ2>0, the ratio η may be selected to
provide an indefinite permeability matrix according to equations
(20)-(21) (moreover, for η substantially equal to
|μ1/μ2|, the indefinite permeability medium is
substantially a degenerate indefinite permeability medium).

[0042]Exemplary planar stacks of alternating materials, providing an
effective continuous medium having an indefinite permittivity matrix,
include an alternating silver/silica layered system described in B. Wood
et al, "Directed subwavelength imaging using a layered medal-dielectric
system," Phys. Rev. B 74, 115116 (2006), and an alternating doped/undoped
semiconductor layered system described in A. J. Hoffman, "Negative
refraction in semiconductor metamaterials," Nature Materials 6, 946
(2007), each of which is herein incorporated by reference. More
generally, materials having a positive permittivity include but are not
limited to: semiconductors (e.g. at frequencies higher than a plasma
frequency of the semiconductor) and insulators such as dielectric
crystals (e.g. silicon oxide, aluminum oxide, calcium fluoride, magnesium
fluoride), glasses, ceramics, and polymers (e.g. photoresist, PMMA).
Generally these materials have a positive permeability as well (which may
be substantially equal to unity if the material is substantially
nonmagnetic). In some embodiments a positive permittivity material is a
gain medium, which may be pumped, for example, to reduce or overcome
other losses such as ohmic losses (cf. an exemplary alternating
silver/gain layered system described in S. Ramakrishna and J. B. Pendry,
"Removal of absorption and increase in resolution in a near-field lens
via optical gain," Phys. Rev. B 67, 201101(R) (2003), herein incorporated
by reference). Examples of gain media include semiconductor laser
materials (e.g. GaN, AlGaAs), doped insulator laser materials (e.g.
rare-earth doped crystals, glasses, or ceramics), and Raman gain
materials. Materials having a negative permeability include but are not
limited to: ferrites, magnetic garnets or spinels, artificial ferrites,
and other ferromagnetic or ferrimagnetic materials (e.g. at frequencies
above a ferromagnetic or ferrimagnetic resonance frequency of the
material; cf. F. J. Rachford, "Tunable negative refractive index
composite," U.S. patent application Ser. No. 11/279/460, herein
incorporated by reference). Materials having a negative permittivity
include but are not limited to: metals (e.g. at frequencies less than a
plasma frequency of the metal) including the noble metals (Cu, Au, Ag);
semiconductors (e.g. at frequencies less than a plasma frequency of the
semiconductor); and polar crystals (e.g. SiC, LiTaO3, LiF, ZnSe) at
frequencies within a restrahlen band of the polar crystal (cf. G.
Schvets, "Photonic approach to making a material with a negative index of
refraction," Phys. Rev. B 67, 035109 (2003) and T. Tauber et al,
"Near-field microscopy through a SiC superlens," Science 313, 1595
(2006), each of which is herein incorporated by reference). For
applications involving semiconductors, the plasma frequency (which may be
regarded as a frequency at which the semiconductor permittivity changes
sign) is related to the density of free carriers within the
semiconductor, and this free carrier density may be controlled in various
ways (e.g. by chemical doping, photodoping, temperature change, carrier
injection, etc.). Thus, for example, a layered system comprising
interleaved layers of a first semiconductor material having a first
plasma frequency and a second semiconductor material having a second
plasma frequency may provide an indefinite permittivity (per equations
(18)-(19)) in a window of frequencies intermediate the first plasma
frequency and the second plasma frequency, and this window may be
controlled, e.g., by differently doping the first and second
semiconductor materials.

[0043]In some applications a layered structure includes a succession of
adjacent layers that are substantially nonplanar. The preceding exemplary
layered structure--consisting of successive planar layers, each layer
having a layer normal direction (the z direction) that is constant along
the transverse extent of the layer and a layer thickness that is constant
along the transverse extent of the layer--may be extended to a nonplanar
layered structure, consisting of successive nonplanar layers, each layer
having a layer normal direction that is non-constant along the transverse
extent of the layer and, optionally, a layer thickness that is
non-constant along the transverse extent of the layer. Some examples of
cylindrical and/or spherical layered structures are described in A.
Salandrino and N. Engheta, "Far-field subdiffraction optical microscopy
using metamaterial crystals: Theory and simulations," Phys. Rev. B 74,
075103 (2006); Z. Jacob et al, "Optical hyperlens: Far-field imaging
beyond the diffraction limit," Opt. Exp. 14, 8247 (2006); Z. Liu et al,
"Far field optical hyperlens magnifying sub-diffraction-limited objects,"
Science 315, 1686 (2007); and H. Lee, "Development of optical hyperlens
for imaging below the diffraction limit," Opt. Exp. 15, 15886 (2007);
each of which is herein incorporated by reference. More generally, for an
alternating nonplanar layered structure, supposing that the layers have
curvature radii substantially less than their respective thicknesses, and
transverse layer thickness gradients substantially less than one, the
nonplanar layered structure may be characterized as an effective
continuous medium having (effective) anisotropic constitutive parameters
as in equations (18)-(21), where the z direction is replaced by a layer
normal direction that may vary with position within the layered
structure, the x direction is replaced by a first transverse direction
perpendicular to the layer normal direction, the y direction is replaced
by a second transverse direction mutually perpendicular to the layer
normal direction and the first transverse direction, and the layer
thickness ratio η=d2/d1 is a ratio of local layer
thicknesses d1 and d2 that may vary with position throughout
the layered structure (so the ratio η may vary with position as
well). The nonplanar layered structure may thus provide an indefinite
medium having a spatially-varying axial direction that corresponds to the
layer normal direction. Suppose, for example, that the spatially-varying
axial direction of an indefinite medium is given by a vector field
uA(r) that is equal to or parallel to a conservative vector field,
i.e.

uA∝∇Φ (22)

for a scalar potential function Φ; then the indefinite medium may be
provided by a nonplanar layered structure where the interfaces of
adjacent layers correspond to equipotential surfaces of the function
Φ.

[0044]Nonplanar layered structures may be fabricated by various methods
that are known to those of skill in the art. In a first example, J. A.
Folta, "Dynamic mask for producing uniform or graded-thickness thin
films," U.S. Pat. No. 7,062,348 (herein incorporated by reference),
describes vapor deposition systems that utilize a moving mask, where the
velocity and position of the moving mask may be computer controlled to
precisely tailor the thickness distributions of deposited films. In a
second example, Tzu-Yu Wang, "Graded thickness optical element and method
of manufacture therefor," U.S. Pat. No. 6,606,199 (herein incorporated by
reference), describes methods for depositing graded thickness layers
through apertures in a masking layer.

[0045]With reference now to FIG. 1, an illustrative embodiment is depicted
that includes a conversion structure 100 with indefinite electromagnetic
parameters, the conversion structure having a first surface region 111
and a second surface region 112, the first surface region and the second
surface region being substantially planar and substantially parallel.
This and other drawings, unless context dictates otherwise, can represent
a planar view of a three-dimensional embodiment, or a two-dimensional
embodiment (e.g. in FIG. 1 where the conversion structure is placed
inside a metallic or dielectric slab waveguide oriented normal to the
page). The conversion structure is responsive to an evanescent
electromagnetic wave (depicted schematically as exponential tails 120) at
the first surface region to convey a propagating electromagnetic wave
(depicted schematically as dashed rays 125) from the first surface region
to the second surface region, and to provide a non-evanescent
electromagnetic wave (depicted schematically as the wavy rays 130) at the
second surface region. In some embodiments the provided non-evanescent
electromagnetic wave is a freely-propagating electromagnetic wave, e.g. a
wave that is transmitted by and freely radiates from the second surface
region (including diverging propagating waves, converging propagating
waves, and substantially planar propagating waves). In other embodiments
the provided non-evanescent wave is a confinedly-propagating
electromagnetic wave, e.g. a wave that is transmitted by the second
surface region into a propagating guided wave mode (as in a waveguide,
transmission line, optical fiber, etc.) While the first and second
surface regions 111 and 112 are depicted in FIG. 1 as exterior surfaces
of the conversion structure 100, in other embodiments the first surface
region and/or the second surface region may be at least partially
interior to the conversion structure (e.g. where the conversion structure
includes one or more of a refractive cladding, an impedance-matching
layer, input or output optical components, etc.). The use of a ray
description, in FIG. 1 and elsewhere, is a heuristic convenience for
purposes of visual illustration, and is not intended to connote any
limitations or assumptions of geometrical optics; further, the elements
depicted in FIG. 1 can have spatial dimensions that are variously less
than, greater than, or comparable to a wavelength of interest. At the
first surface region 111, the evanescent electromagnetic wave 120 may be
characterized by a first transverse wavevector kT.sup.(1)
(corresponding to a surface parallel direction of the first surface
region indicated as the vectors 141 in FIG. 1) that exceeds a first
maximum transverse wavevector kmax.sup.(1) for non-evanescent waves
(cf. equation (2) and related preceding text):

k T ( 1 ) > k max ( 1 ) = 2 π n 1 f c
= 2 π f v 1 ( 23 ) ##EQU00011##

where f is the frequency of the evanescent electromagnetic wave and
v1 is a phase velocity (at the frequency f) for electromagnetic
waves in a first region outside the conversion structure 100 and adjacent
to the first surface region 111 (the phase velocity may correspond to an
index of refraction n1 for a refractive medium, possibly vacuum, in
the first region, according to the relation v1=c/n). At the second
surface region 112, the non-evanescent electromagnetic wave 130 may be
characterized by a second transverse wavevector kT.sup.(2)
(corresponding to a surface parallel direction of the second surface
region indicated as the vectors 142 in FIG. 1) that does not exceed a
second maximum transverse wavevector kmax.sup.(2) for non-evanescent
waves (cf. equation (2) and related preceding text):

k T ( 2 ) > k max ( 2 ) = 2 π n 2 f c
= 2 π f v 2 ( 24 ) ##EQU00012##

where f is the frequency of the non-evanescent electromagnetic wave and
v2 is a phase velocity (at the frequency f) for electromagnetic
waves in a second region outside the conversion structure 100 and
adjacent to the second surface region 112 (the phase velocity may
correspond to an index of refraction n2 for a refractive medium,
possibly vacuum, in the second region, according to the relation
v2=c/n2, where n2 may be equal to or different than
n1).

[0046]In the illustrative embodiment of FIG. 1, the conversion structure
100 has indefinite electromagnetic parameters, i.e. the conversion
structure provides an indefinite medium (i.e. an electromagnetic medium
having an indefinite permittivity and/or an indefinite permeability, as
discussed above) that is responsive to the evanescent electromagnetic
wave 120 to convey a propagating electromagnetic wave from the first
surface region 111 to the second surface region 112. The indefinite
medium defines an axial direction (indicated by the vectors 150 at
various positions within the indefinite medium), which, as previously
discussed, corresponds to a first eigenvector of the indefinite
permittivity matrix and/or the indefinite permeability matrix; and the
indefinite medium further defines a transverse direction (indicated by
the vectors 151 at various positions within the indefinite medium) that
is perpendicular to the axial direction and corresponds to a second
eigenvector of the indefinite permittivity matrix and/or the indefinite
permeability matrix. In the illustrative embodiment of FIG. 1, the axial
direction 150 is a non-constant axial direction that is a function of
location within the conversion structure 100, i.e. the axial direction
may be regarded as a vector field (a vector-valued function of location).
Moreover, the axial direction is generally directed from the first
surface region 111 to the second surface region 112, i.e. axial field
lines corresponding to the axial direction vector field extend from the
first surface region to the second surface region. In FIG. 1, the dashed
rays 125, indicating the propagating electromagnetic wave, also
correspond to axial field lines, because the illustrative embodiment
depicts a degenerate indefinite medium, i.e. an indefinite medium, as
described previously, that substantially conveys electromagnetic energy
along a propagation direction that corresponds to the axial direction of
the indefinite medium. (This depiction is not intended to be limiting: in
other embodiments, the indefinite medium is a "non-degenerate" indefinite
medium that substantially conveys electromagnetic energy along multiple
propagation directions--e.g. along at least two propagation directions,
each of the at least two directions having a substantially common angle
with respect to the axial direction, or along a plurality of propagation
directions, the plurality of propagation directions substantially
composing a cone having a cone axis that substantially coincides with the
axial direction.)

[0047]Referring again to FIG. 1, the propagating electromagnetic field 125
may be characterized by a transverse wavevector kT that corresponds
to the transverse direction 151. In the present example, the axial field
lines (corresponding to the vector field that describes the axial
direction 150) diverge geometrically as they proceed from the first
surface region 111 to the second surface region 112, and this geometric
divergence may provide a substantially continuous variation of the
transverse wavevector kT, from a first transverse wavevector
kT.sup.(1) at the first surface region (as in equation (23), to
match the transverse wavevector of the evanescent electromagnetic wave
120) to a second transverse wavevector kT.sup.(2) at the second
surface region (as in equation (24), to match the transverse wavevector
of the non-evanescent electromagnetic wave 130). Thus, the geometric
divergence of the axial field lines admits the conversion of an
evanescent electromagnetic wave 120 to a non-evanescent electromagnetic
wave 130, by supporting a propagating electromagnetic wave 125 having a
substantially continuous variation of transverse wavevector from an
initial transverse wavevector that exceeds a maximum wavevector for
non-evanescent waves to a final transverse wavevector that does not
exceed a maximum wavevector for non-evanescent waves.

[0048]With reference now to FIGS. 2 and 3, layered structures are depicted
as exemplary implementations of the conversion structure 100 of FIG. 1.
In the exemplary implementations of FIGS. 2 and 3, the conversion
structure 100 includes (as above) a first surface region 111 and a second
surface region 112 that are substantially planar and substantially
parallel; intermediate the first and second surface regions, a layered
structure provides an effective continuous medium that corresponds to an
indefinite medium. The layered structure includes layers of a first
material 201 interleaved with layers of a second material 202, where the
first and second materials have electromagnetic parameters (e.g.
permittivities and/or permeabilities) that are oppositely-signed, as
described previously. In the exemplary implementations of FIGS. 2 and 3,
the alternating layers 201 and 202 are substantially nonplanar, having a
layer normal direction that varies with position throughout the layered
structure (i.e. from layer to layer and/or along the transverse extent of
each layer), and this layer normal direction corresponds to the axial
direction (as depicted by the vectors 150 in FIG. 1) of the provided
indefinite medium (equivalently, regarding the interfaces between
alternating layers 201 and 202 as equipotential surfaces of a scalar
function Φ, the gradient of Φ is locally parallel to the axial
direction 150 as per equation (22)). In the exemplary implementation of
FIG. 3, a first layer 301 of the layered structure substantially
coincides with the first surface region 111, and a last layer 302 of the
layered structure substantially coincides with the second surface region
112, but this is not intended to be limiting (e.g. in the exemplary
implementation of FIG. 2, neither the first surface region 111 nor the
second surface region 112 substantially coincides with a layer of the
layered structure). The nonplanar alternating layers may have
substantially uniform thickness throughout the transverse extents of the
layers, as in FIG. 2; or substantially non-uniform thicknesses throughout
the transverse extents of the layers, as in FIG. 3; or a combination
thereof.

[0049]With reference now to FIG. 4, an illustrative embodiment is depicted
that includes a conversion structure 100 with indefinite electromagnetic
parameters, the conversion structure having a first surface region 111
that is substantially planar and a second surface region 112 that is
substantially nonplanar. In the illustrative embodiment, the
substantially nonplanar second surface region 112 is depicted as a convex
surface region (i.e. the configuration is a "plano-convex"
configuration), but this is an exemplary configuration and is not
intended to be limiting: other embodiments (not depicted) provide a
substantially nonplanar second surface region 112 that is concave (a
"plano-concave" configuration), or that includes a first subregion that
is concave and a second subregion that is convex. This and other
drawings, unless context dictates otherwise, can represent a planar view
of a three-dimensional embodiment, or a two-dimensional embodiment (e.g.
in FIG. 4 where the conversion structure is placed inside a metallic or
dielectric slab waveguide oriented normal to the page). The conversion
structure is responsive to an evanescent electromagnetic wave (depicted
schematically as exponential tails 120) at the first surface region to
convey a propagating electromagnetic wave (depicted schematically as
dashed rays 125) from the first surface region to the second surface
region, and to provide a non-evanescent electromagnetic wave (depicted
schematically as the wavy rays 130) at the second surface region. In some
embodiments the provided non-evanescent electromagnetic wave is a
freely-propagating electromagnetic wave, e.g. a wave that is transmitted
by and freely radiates from the second electromagnetic surface (including
diverging propagating waves, converging propagating waves, and
substantially planar propagating waves). In other embodiments the
provided non-evanescent wave is a confinedly-propagating electromagnetic
wave, e.g. a wave that is transmitted by the second electromagnetic
surface into a propagating guided wave mode (as in a waveguide,
transmission line, optical fiber, etc.) While the first and second
surface regions 111 and 112 are depicted in FIG. 4 as exterior surfaces
of the conversion structure 100, in other embodiments the first surface
region and/or the second surface region may be at least partially
interior to the conversion structure (e.g. where the conversion structure
includes one or more of a refractive cladding, an impedance-matching
layer, input or output optical components, etc.). The use of a ray
description, in FIG. 4 and elsewhere, is a heuristic convenience for
purposes of visual illustration, and is not intended to connote any
limitations or assumptions of geometrical optics; further, the elements
depicted in FIG. 4 can have spatial dimensions that are variously less
than, greater than, or comparable to a wavelength of interest. At the
first surface region 111, the evanescent electromagnetic wave 120 may be
characterized by a first transverse wavevector kT.sup.(1)
(corresponding to a surface parallel direction of the first surface
region indicated as the vectors 141 in FIG. 4) that exceeds a first
maximum transverse wavevector kmax.sup.(1) defined as in equation
(23). At the second surface region 112, the non-evanescent
electromagnetic wave 130 may be characterized by a second transverse
wavevector kT.sup.(2) (corresponding to a surface parallel direction
of the second surface region indicated as the vectors 142 in FIG. 4) that
does not exceed a second maximum transverse wavevector kmax.sup.(2)
defined as in equation (24).

[0050]In the illustrative embodiment of FIG. 4, the conversion structure
100 has indefinite electromagnetic parameters, i.e. the conversion
structure provides an indefinite medium (i.e. an electromagnetic medium
having an indefinite permittivity and/or an indefinite permeability, as
discussed above) that is responsive to the evanescent electromagnetic
wave 120 to convey a propagating electromagnetic wave from the first
surface region 111 to the second surface region 112. The indefinite
medium defines an axial direction (indicated by the vectors 150 at
various positions within the indefinite medium), which, as previously
discussed, corresponds to a first eigenvector of the indefinite
permittivity matrix and/or the indefinite permeability matrix; and the
indefinite medium further defines a transverse direction (indicated by
the vectors 151 at various positions within the indefinite medium) that
is perpendicular to the axial direction and corresponds to a second
eigenvector of the indefinite permittivity matrix and/or the indefinite
permeability matrix. In the illustrative embodiment of FIG. 4, the axial
direction 150 is a non-constant axial direction that is a function of
location within the conversion structure 100, i.e. the axial direction
may be regarded as a vector field (a vector-valued function of location).
Moreover, the axial direction is generally directed from the first
surface region 111 to the second surface region 112, i.e. axial field
lines corresponding to the axial direction vector field extend from the
first surface region to the second surface region. In FIG. 4, the dashed
rays 125, indicating the propagating electromagnetic wave, also
correspond to axial field lines, because the illustrative embodiment
depicts a degenerate indefinite medium, i.e. an indefinite medium, as
described previously, that substantially conveys electromagnetic energy
along a propagation direction that corresponds to the axial direction of
the indefinite medium. (This depiction is not intended to be limiting: in
other embodiments, the indefinite medium is a "non-degenerate" indefinite
medium that substantially conveys electromagnetic energy along multiple
propagation directions--e.g. along at least two propagation directions,
each of the at least two directions having a substantially common angle
with respect to the axial direction, or along a plurality of propagation
directions, the plurality of propagation directions substantially
composing a cone having a cone axis that substantially coincides with the
axial direction.)

[0051]Referring again to FIG. 4, the propagating electromagnetic field 125
may be characterized by a transverse wavevector kT that corresponds
to the transverse direction 151. In the present example, the axial field
lines (corresponding to the vector field that describes the axial
direction 150) diverge geometrically as they proceed from the first
surface region 111 to the second surface region 112, and this geometric
divergence may provide a substantially continuous variation of the
transverse wavevector kT, from a first transverse wavevector
kT.sup.(1) at the first surface region (as in equation (23), to
match the transverse wavevector of the evanescent electromagnetic wave
120) to a second transverse wavevector kT.sup.(2) at the second
surface region(as in equation (24), to match the transverse wavevector of
the non-evanescent electromagnetic wave 130). Thus, the geometric
divergence of the axial field lines admits the conversion of an
evanescent electromagnetic wave 120 to a non-evanescent electromagnetic
wave 130, by supporting a propagating electromagnetic wave 125 having a
substantially continuous variation of transverse wavevector from an
initial transverse wavevector that exceeds a maximum wavevector for
non-evanescent waves to a final transverse wavevector that does not
exceed a maximum wavevector for non-evanescent waves.

[0052]With reference now to FIGS. 5 and 6, layered structures are depicted
as exemplary implementations of the conversion structure 100 of FIG. 4.
In the exemplary implementations of FIGS. 5 and 6, the conversion
structure 100 includes (as in FIG. 4) a first surface region 111 that is
substantially planar and a second surface region 112 that substantially
nonplanar; intermediate the first and second surface regions, a layered
structure provides an effective continuous medium that corresponds to an
indefinite medium. The layered structure includes layers of a first
material 201 interleaved with layers of a second material 202, where the
first and second materials have electromagnetic parameters (e.g.
permittivities and/or permeabilities) that are oppositely-signed, as
described previously. In the exemplary implementations of FIGS. 5 and 6,
the alternating layers 201 and 202 are substantially nonplanar, having a
layer normal direction that varies with position throughout the layered
structure (i.e. from layer to layer and/or along the transverse extent of
each layer), and this layer normal direction corresponds to the axial
direction (as depicted by the vectors 150 in FIG. 4) of the provided
indefinite medium (equivalently, regarding the interfaces between
alternating layers 201 and 202 as equipotential surfaces of a scalar
function Φ, the gradient of Φ is locally parallel to the axial
direction 150 as per equation (22)). In the exemplary implementation of
FIG. 6, a first layer 301 of the layered structure substantially
coincides with the first surface region 111, and a last layer 302 of the
layered structure substantially coincides with the second surface region
112, but this is not intended to be limiting (e.g. in the exemplary
implementation of FIG. 5, only the second surface region 112
substantially coincides with a layer 302 of the layered structure). The
nonplanar alternating layers may have substantially uniform thickness
throughout the transverse extents of the layers, as in FIG. 5; or
substantially non-uniform thicknesses throughout the transverse extents
of the layers, as in FIG. 6; or a combination thereof.

[0053]With reference now to FIG. 7, an illustrative embodiment is depicted
that includes a conversion structure 100 with indefinite electromagnetic
parameters, the conversion structure having a first surface region 111
that is substantially nonplanar and a second surface region 112 that is
substantially planar. In the illustrative embodiment, the substantially
nonplanar first surface region 111 is depicted as a concave surface
region (i.e. the configuration is a "concave-plano" configuration), but
this is an exemplary configuration and is not intended to be limiting:
other embodiments (not depicted) provide a substantially nonplanar first
surface region 111 that is convex (a "convex-plano" configuration), or
that includes a first subregion that is concave and a second subregion
that is convex. This and other drawings, unless context dictates
otherwise, can represent a planar view of a three-dimensional embodiment,
or a two-dimensional embodiment (e.g. in FIG. 7 where the conversion
structure is placed inside a metallic or dielectric slab waveguide
oriented normal to the page). The conversion structure is responsive to
an evanescent electromagnetic wave (depicted schematically as exponential
tails 120) at the first surface region to convey a propagating
electromagnetic wave (depicted schematically as dashed rays 125) from the
first surface region to the second surface region, and to provide a
non-evanescent electromagnetic wave (depicted schematically as the wavy
rays 130) at the second surface region. In some embodiments the provided
non-evanescent electromagnetic wave is a freely-propagating
electromagnetic wave, e.g. a wave that is transmitted by and freely
radiates from the second electromagnetic surface (including diverging
propagating waves, converging propagating waves, and substantially planar
propagating waves). In other embodiments the provided non-evanescent wave
is a confinedly-propagating electromagnetic wave, e.g. a wave that is
transmitted by the second electromagnetic surface into a propagating
guided wave mode (as in a waveguide, transmission line, optical fiber,
etc.) While the first and second surface regions 111 and 112 are depicted
in FIG. 7 as exterior surfaces of the conversion structure 100, in other
embodiments the first surface region and/or the second surface region may
be at least partially interior to the conversion structure (e.g. where
the conversion structure includes one or more of a refractive cladding,
an impedance-matching layer, input or output optical components, etc.).
The use of a ray description, in FIG. 7 and elsewhere, is a heuristic
convenience for purposes of visual illustration, and is not intended to
connote any limitations or assumptions of geometrical optics; further,
the elements depicted in FIG. 7 can have spatial dimensions that are
variously less than, greater than, or comparable to a wavelength of
interest. At the first surface region 111, the evanescent electromagnetic
wave 120 may be characterized by a first transverse wavevector
kT.sup.(1) (corresponding to a surface parallel direction of the
first surface region indicated as the vectors 141 in FIG. 7) that exceeds
a first maximum transverse wavevector kmax.sup.(1) defined as in
equation (23). At the second surface region 112, the non-evanescent
electromagnetic wave 130 may be characterized by a second transverse
wavevector kT.sup.(2) (corresponding to a surface parallel direction
of the second surface region indicated as the vectors 142 in FIG. 7) that
does not exceed a second maximum transverse wavevector kmax.sup.(2)
defined as in equation (24).

[0054]In the illustrative embodiment of FIG. 7, the conversion structure
100 has indefinite electromagnetic parameters, i.e. the conversion
structure provides an indefinite medium (i.e. an electromagnetic medium
having an indefinite permittivity and/or an indefinite permeability, as
discussed above) that is responsive to the evanescent electromagnetic
wave 120 to convey a propagating electromagnetic wave from the first
surface region 111 to the second surface region 112. The indefinite
medium defines an axial direction (indicated by the vectors 150 at
various positions within the indefinite medium), which, as previously
discussed, corresponds to a first eigenvector of the indefinite
permittivity matrix and/or the indefinite permeability matrix; and the
indefinite medium further defines a transverse direction (indicated by
the vectors 151 at various positions within the indefinite medium) that
is perpendicular to the axial direction and corresponds to a second
eigenvector of the indefinite permittivity matrix and/or the indefinite
permeability matrix. In the illustrative embodiment of FIG. 7, the axial
direction 150 is a non-constant axial direction that is a function of
location within the conversion structure 100, i.e. the axial direction
may be regarded as a vector field (a vector-valued function of location).
Moreover, the axial direction is generally directed from the first
surface region 111 to the second surface region 112, i.e. axial field
lines corresponding to the axial direction vector field extend from the
first surface region to the second surface region. In FIG. 7, the dashed
rays 125, indicating the propagating electromagnetic wave, also
correspond to axial field lines, because the illustrative embodiment
depicts a degenerate indefinite medium, i.e. an indefinite medium, as
described previously, that substantially conveys electromagnetic energy
along a propagation direction that corresponds to the axial direction of
the indefinite medium. (This depiction is not intended to be limiting: in
other embodiments, the indefinite medium is a "non-degenerate" indefinite
medium that substantially conveys electromagnetic energy along multiple
propagation directions--e.g. along at least two propagation directions,
each of the at least two directions having a substantially common angle
with respect to the axial direction, or along a plurality of propagation
directions, the plurality of propagation directions substantially
composing a cone having a cone axis that substantially coincides with the
axial direction.)

[0055]Referring again to FIG. 7, the propagating electromagnetic field 125
may be characterized by a transverse wavevector kT that corresponds
to the transverse direction 151. In the present example, the axial field
lines (corresponding to the vector field that describes the axial
direction 150) diverge geometrically as they proceed from the first
surface region 111 to the second surface region 112, and this geometric
divergence may provide a substantially continuous variation of the
transverse wavevector kT, from a first transverse wavevector
kT.sup.(1) at the first surface region (as in equation (23), to
match the transverse wavevector of the evanescent electromagnetic wave
120) to a second transverse wavevector kT.sup.(2) at the second
surface region (as in equation (24), to match the transverse wavevector
of the non-evanescent electromagnetic wave 130). Thus, the geometric
divergence of the axial field lines admits the conversion of an
evanescent electromagnetic wave 120 to a non-evanescent electromagnetic
wave 130, by supporting a propagating electromagnetic wave 125 having a
substantially continuous variation of transverse wavevector from an
initial transverse wavevector that exceeds a maximum wavevector for
non-evanescent waves to a final transverse wavevector that does not
exceed a maximum wavevector for non-evanescent waves.

[0056]With reference now to FIGS. 8 and 9, layered structures are depicted
as exemplary implementations of the conversion structure 100 of FIG. 7.
In the exemplary implementations of FIGS. 8 and 8, the conversion
structure 100 includes (as in FIG. 7) a first surface region 111 that is
substantially nonplanar and a second surface region 112 that
substantially planar; intermediate the first and second surface regions,
a layered structure provides an effective continuous medium that
corresponds to an indefinite medium. The layered structure includes
layers of a first material 201 interleaved with layers of a second
material 202, where the first and second materials have electromagnetic
parameters (e.g. permittivities and/or permeabilities) that are
oppositely-signed, as described previously. In the exemplary
implementations of FIGS. 8 and 9, the alternating layers 201 and 202 are
substantially nonplanar, having a layer normal direction that varies with
position throughout the layered structure (i.e. from layer to layer
and/or along the transverse extent of each layer), and this layer normal
direction corresponds to the axial direction (as depicted by the vectors
150 in FIG. 7) of the provided indefinite medium (equivalently, regarding
the interfaces between alternating layers 201 and 202 as equipotential
surfaces of a scalar function Φ, the gradient of Φ is locally
parallel to the axial direction 150 as per equation (22)). In the
exemplary implementation of FIG. 9, a first layer 301 of the layered
structure substantially coincides with the first surface region 111, and
a last layer 302 of the layered structure substantially coincides with
the second surface region 112, but this is not intended to be limiting
(e.g. in the exemplary implementation of FIG. 8, only the first surface
region 112 substantially coincides with a layer 301 of the layered
structure). The nonplanar alternating layers may have substantially
uniform thickness throughout the transverse extents of the layers, as in
FIG. 8; or substantially non-uniform thicknesses throughout the
transverse extents of the layers, as in FIG. 9; or a combination thereof.

[0057]With reference now to FIG. 10, various illustrative embodiments are
depicted that include a conversion structure 100 with indefinite
electromagnetic parameters, the conversion structure having a first
surface region 111 and a second surface region 112, the first surface
region and the second surface region being substantially nonplanar and
substantially non-concentric. In general, a first surface region and a
second surface region are non-concentric if: the first surface region has
a non-constant curvature, and/or the second surface region has a
non-constant curvature, and/or a center of an osculating circle of the
first surface region is different than a center of an osculating circle
of the second surface region (where the osculating circles are coplanar).
Embodiment 1001 depicts a conversion structure 100 having a first surface
region 111 and a second surface region 112 that are both convex (a
"bi-convex" configuration). Embodiment 1002 depicts a conversion
structure 100 having a first surface region 111 and a second surface
region 112 that are both concave (a "bi-concave" configuration).
Embodiment 1003 depicts a conversion structure 100 having a first surface
region 111 that is concave and a second surface region 112 that is
convex, where a center of curvature of the first surface region is to the
right of--i.e. nearer to the conversion structure than--a center of
curvature of the second surface region, i.e. a concave-convex "negative
meniscus" configuration (another exemplary embodiment--not
shown--provides a concave-convex "positive meniscus" configuration where
the center of curvature of the first surface region is to the left--i.e.
farther from the conversion structure than--the center of curvature of
the second surface region). Embodiment 1004 depicts a conversion
structure 100 having a first surface region 111 that is convex and a
second surface region 112 that is concave, where a center of curvature of
the first surface region is to the left of--i.e. nearer to the conversion
structure than--a center of curvature of the second surface region, i.e.
a convex-concave "positive meniscus" configuration (another exemplary
embodiment--not shown--provides a convex-concave "negative meniscus"
configuration where the center of curvature of the first surface region
is to the right of--i.e. farther from the conversion structure than--the
center of curvature of the second surface region). Embodiment 1005
depicts a conversion structure having a first surface region 111 that is
partially convex and partially concave and a second surface region 112
that is partially convex and partially concave (in other embodiments, not
shown, the first surface region 111 is convex only or concave only and
the second surface region 112 is partially convex and partially concave,
or the first surface region 111 is partially convex and partially concave
and the second surface region 112 is convex only or concave only). As
elsewhere, the depictions in FIG. 10 can represent planar views of
three-dimensional embodiments, or a two-dimensional embodiment (e.g.
where the conversion structure 100 is placed inside a metallic or
dielectric slab waveguide oriented normal to the page). In each
embodiment of FIG. 10, the conversion structure 100 is responsive to an
evanescent electromagnetic wave (depicted schematically as exponential
tails 120) at the first surface region 111 to convey a propagating
electromagnetic wave (depicted schematically as dashed rays 125) from the
first surface region to the second surface region, and to provide a
non-evanescent electromagnetic wave (depicted schematically as the wavy
rays 130) at the second surface region 112. In some applications the
provided non-evanescent electromagnetic wave is a freely-propagating
electromagnetic wave, e.g. a wave that is transmitted by and freely
radiates from the second electromagnetic surface (including diverging
propagating waves, converging propagating waves, and substantially planar
propagating waves). In other applications the provided non-evanescent
wave is a confinedly-propagating electromagnetic wave, e.g. a wave that
is transmitted by the second electromagnetic surface into a propagating
guided wave mode (as in a waveguide, transmission line, optical fiber,
etc.) While the first and second surface regions 111 and 112 are depicted
in FIG. 10 as exterior surfaces of the conversion structure 100, in other
embodiments the first surface region and/or the second surface region may
be at least partially interior to the conversion structure (e.g. where
the conversion structure includes one or more of a refractive cladding,
an impedance-matching layer, input or output optical components, etc.).
The use of a ray description, in FIG. 10 and elsewhere, is a heuristic
convenience for purposes of visual illustration, and is not intended to
connote any limitations or assumptions of geometrical optics; further,
the elements depicted in FIG. 10 can have spatial dimensions that are
variously less than, greater than, or comparable to a wavelength of
interest. At the first surface region 111, the evanescent electromagnetic
wave 120 may be characterized by a first transverse wavevector
kT.sup.(1) (corresponding to a surface parallel direction of the
first surface region--for simplicity this surface parallel direction of
the first surface region is not depicted in the embodiments of FIG. 10,
but should be apparent from the analogous elements 141 depicted in FIGS.
1, 4, and 7) that exceeds a first maximum transverse wavevector
kmax.sup.(1) defined as in equation (23). At the second surface
region 112, the non-evanescent electromagnetic wave 130 may be
characterized by a second transverse wavevector kT.sup.(2)
(corresponding to a surface parallel direction of the second surface
region--again for simplicity this surface parallel direction of the
second surface region is not depicted in the embodiments of FIG. 10, but
should be apparent from the analogous elements 142 depicted in FIGS. 1,
4, and 7) that does not exceed a second maximum transverse wavevector
kmax.sup.(2) defined as in equation (24).

[0058]In the illustrative embodiments of FIG. 10, the conversion structure
100 has indefinite electromagnetic parameters, i.e. the conversion
structure provides an indefinite medium (i.e. an electromagnetic medium
having an indefinite permittivity and/or an indefinite permeability, as
discussed above) that is responsive to the evanescent electromagnetic
wave 120 to convey a propagating electromagnetic wave from the first
surface region 111 to the second surface region 112. The indefinite
medium defines an axial direction (again for simplicity this axial
direction is not depicted in the embodiments of FIG. 10, but should be
apparent from the analogous elements 150 depicted in FIGS. 1, 4, and 7),
which, as previously discussed, corresponds to a first eigenvector of the
indefinite permittivity matrix and/or the indefinite permeability matrix;
and the indefinite medium further defines a transverse direction (again
for simplicity this transverse direction is not depicted in the
embodiments of FIG. 10, but should be apparent from the analogous
elements 151 depicted in FIGS. 1, 4, and 7) that is perpendicular to the
axial direction and corresponds to a second eigenvector of the indefinite
permittivity matrix and/or the indefinite permeability matrix. In the
illustrative embodiments of FIG. 10, the axial direction 150 is a
non-constant axial direction that is a function of location within the
conversion structure 100, i.e. the axial direction may be regarded as a
vector field (a vector-valued function of location). Moreover, the axial
direction is generally directed from the first surface region 111 to the
second surface region 112, i.e. axial field lines corresponding to the
axial direction vector field extend from the first surface region to the
second surface region. In FIG. 1, the dashed rays 125, indicating the
propagating electromagnetic wave, also correspond to axial field lines,
because the illustrative embodiment depicts a degenerate indefinite
medium, i.e. an indefinite medium, as described previously, that
substantially conveys electromagnetic energy along a propagation
direction that corresponds to the axial direction of the indefinite
medium. (This depiction is not intended to be limiting: in other
embodiments, the indefinite medium is a "non-degenerate" indefinite
medium that substantially conveys electromagnetic energy along multiple
propagation directions--e.g. along at least two propagation directions,
each of the at least two directions having a substantially common angle
with respect to the axial direction, or along a plurality of propagation
directions, the plurality of propagation directions substantially
composing a cone having a cone axis that substantially coincides with the
axial direction.)

[0059]Referring again to FIG. 10, the propagating electromagnetic field
125 may be characterized by a transverse wavevector kT that
corresponds to the transverse direction (again for simplicity this
transverse direction is not depicted in the embodiments of FIG. 10, but
should be apparent from the analogous elements 151 depicted in FIGS. 1,
4, and 7). In the present example, the axial field lines (corresponding
to the vector field that describes the axial direction) diverge
geometrically as they proceed from the first surface region 111 to the
second surface region 112, and this geometric divergence may provide a
substantially continuous variation of the transverse wavevector kT,
from a first transverse wavevector kT.sup.(1) at the first surface
region (as in equation (23), to match the transverse wavevector of the
evanescent electromagnetic wave 120) to a second transverse wavevector
kT.sup.(2) at the second surface region (as in equation (24), to
match the transverse wavevector of the non-evanescent electromagnetic
wave 130). Thus, the geometric divergence of the axial field lines admits
the conversion of an evanescent electromagnetic wave 120 to a
non-evanescent electromagnetic wave 130, by supporting a propagating
electromagnetic wave 125 having a substantially continuous variation of
transverse wavevector from an initial transverse wavevector that exceeds
a maximum wavevector for non-evanescent waves to a final transverse
wavevector that does not exceed a maximum wavevector for non-evanescent
waves.

[0060]With reference now to FIG. 11, a layered structure is depicted as an
exemplary implementation of a conversion structure 100 as in FIG. 10. In
the exemplary implementation of FIG. 11, the conversion structure 100
includes a first surface region 111 and a second surface region 112 that
are substantially nonplanar and substantially non-concentric;
intermediate the first and second surface regions, a layered structure
provides an effective continuous medium that corresponds to an indefinite
medium. The layered structure includes layers of a first material 201
interleaved with layers of a second material 202, where the first and
second materials have electromagnetic parameters (e.g. permittivities
and/or permeabilities) that are oppositely-signed, as described
previously. In the exemplary implementations of FIG. 11, the alternating
layers 201 and 202 are substantially nonplanar, having a layer normal
direction that varies with position throughout the layered structure
(i.e. from layer to layer and/or along the transverse extent of each
layer), and this layer normal direction corresponds to the axial
direction of the provided indefinite medium (equivalently, regarding the
interfaces between alternating layers 201 and 202 as equipotential
surfaces of a scalar function Φ, the gradient of Φ is locally
parallel to the axial direction as per equation (22)). In the exemplary
implementation of FIG. 3, a first layer 301 of the layered structure
substantially coincides with the first surface region 111, and a last
layer 302 of the layered structure substantially coincides with the
second surface region 112, but this is not intended to be limiting (in
other embodiments, not depicted, the first surface region 111 does not
coincide with a layer of the layered structure, and/or the second surface
region 112 does not coincide with a layer of the layered structure). The
nonplanar alternating layers may have substantially uniform thickness
throughout the transverse extents of the layers; or substantially
non-uniform thicknesses throughout the transverse extents of the layers;
or a combination thereof, as in FIG. 11.

[0061]In some embodiments a conversion structure, such as those depicted
in FIGS. 1-11, includes an indefinite transformation medium, i.e. a
transformation medium that has indefinite electromagnetic parameters. For
example, the geometric divergence of the axial field lines, along the
dashed rays 125 in FIGS. 1, 4, 7, and 10, may accord with a coordinate
transformation, e.g. from an untransformed coordinate space in which the
axial field lines do not have a geometric divergence. Recalling the
previous exemplary planar slab of indefinite medium (i.e. as described by
equations (3), (5), (9), and/or (11) and the text accompanying these
equations), the planar slab has an axial direction that corresponds to
the z-axis; thus, axial field lines for the planar slab are straight
lines parallel to the z-axis and perpendicular to the faces of the slab
(for purposes of illustration, suppose that the faces of the slab--its
first and second surface regions--are located at z=0 and z=d,
respectively). To obtain an indefinite transformation medium, suppose
that this planar slab is regarded as the original untransformed medium in
equations (15) and (16), and consider an exemplary coordinate
transformation that maps a portion of the surface z=0 to the first
surface region 111 of the conversion structure 100, and a portion of the
surface z=d to the second surface region 112 of the conversion structure
100, with portions of the intermediate surfaces of constant z (i.e. for
0<z<d) mapped to successive surfaces in the transformed coordinate
space, so that the family of successive surfaces {|0<z<d} spans a
region intermediate the first surface region and the second surface
region in the transformed coordinate space. Further, suppose that the
exemplary coordinate transformation provides an increasing magnification
of the successive constant-z surfaces according to a magnification factor
m(z) so that, for example, two lines that are parallel to the z-axis in
the untransformed coordinate space shall diverge in the transformed
coordinate space, with a geodesic distance between the two lines being
proportional to m(z) on the surface z. For the conversion structure
100 of FIG. 1, having first and second surface regions 111 and 112 that
are substantially parallel and substantially planar, an exemplary
coordinate transformation maps planes of constant z to planes of constant
z' according to the equations

x'=m(z)x

y'=m(z)y

z'=z (25)

where m(z) is a magnification factor that increases with z (e.g. from m=1
at the first surface region to m=M>1 at the second surface region). In
this example, the first and second surface regions of the indefinite
transformation medium, at z'=0 and z'=d, respectively, correspond to the
first and second surface regions 111 and 112 of the conversion structure
100 in FIG. 1. Constitutive parameters of the indefinite transformation
medium (obtained from equations (15) and (16) with the Jacobian matrix
(17) corresponding to coordinate transformation (25)) provide an
indefinite medium with axial field lines that diverge geometrically as
they proceed from the first surface region to the second surface region,
in accordance with the magnifying coordinate transformation (25). Then,
for a propagating electromagnetic wave 125 characterized by a transverse
wavevector kT, the transverse wavevector varies in inverse
proportion to the magnification factor m(z') as the propagating
electromagnetic wave advances from the first surface region 111 to the
second surface region 112, implying the connection

between the first transverse wavevector kT.sup.(1) for the evanescent
electromagnetic wave 120 (at the first surface region) and the second
transverse wavevector kT.sup.(2) for the non-evanescent
electromagnetic wave 130 (at the second surface region). Therefore the
conversion structure 100 will convert an evanescent electromagnetic wave
120 to a non-evanescent electromagnetic wave 130 for a range of
transverse wavevectors kT.sup.(1).di-elect cons.(kmax.sup.(1),
Mkmax.sup.(2)) (cf. equations (23) and (24)); or, reciprocally, the
conversion structure 100 will convert a non-evanescent electromagnetic
wave 130 to an evanescent electromagnetic wave 120 for a range of
transverse wavevectors kT.sup.(2).di-elect
cons.(M-1kmax.sup.(1), kmax.sup.(2)).

[0062]In some embodiments, the planar slab of untransformed indefinite
medium is a degenerate indefinite medium, i.e. providing degenerate
propagation for TM-polarized waves (with |εx| and/or
|εy| substantially less than |εz|),
TE-polarized waves (with |μx| and/or |μy| substantially
less than |μz|), or both. For example, the planar slab may have a
permittivity matrix

= ( x 0 0 0 y 0 0 0 z )
≈ ( 0 0 0 0 0 0 0 0 z ) ( 27
) ##EQU00014##

(where the symbol "≈" indicates that the transverse components are
approximated as zero). In the transformed coordinate space, the new
permittivity tensor is

(in the coordinate basis (x', y', z')), where the latter proportionality
is obtained by substituting equation (25). In some embodiments this
transformation medium may be implemented as a nonplanar layered structure
(cf. the preceding discussion of layered structures), by relating the
vector field (30) to a scalar potential Φ according to equation (22)
whereby the interfaces of adjacent layers in the nonplanar layered
structure correspond to equipotential surfaces of the function Φ. In
a first example, the magnification factor may increase linearly with z,
e.g.

m ( z ) = 1 + ( M - 1 ) z d ; ( 31 )
##EQU00017##

the resultant axial vector field (30) corresponds to a scalar potential
Φ having equipotential surfaces that are concentric spheres (or
cylinders, in a two-dimensional embodiment) centered at z'=-d/(M-1). The
layered structure of FIG. 2 resembles a configuration of this sort;
moreover the layered structures of FIGS. 5 and 8 resemble the
configuration of FIG. 2, absent selected layers so as to have either a
nonplanar first surface region or a nonplanar second surface region of
the conversion structure, but providing similar indefinite medium
properties within the interior of the conversion structure. In a second
example, the magnification factor may increase nonlinearly with z, e.g.

m ( z ) = 1 + M - 1 2 ( 1 - cos π z d
) ( 32 ) ##EQU00018##

(the functional dependence being selected to have m'(0)=m'(d)=0); the
resultant axial vector field (30) corresponds to a scalar potential Φ
having successive equipotential surfaces that evolve from a planar
surface at z'=0 through a series of curved surfaces to another planar
surface at z'=d. The layered structure of FIG. 3 resembles a
configuration of this sort; moreover the layered structures of FIGS. 6
and 9 resemble the configuration of FIG. 3, absent selected layers so as
to have either a nonplanar first surface region or a nonplanar second
surface region of the conversion structure, but providing similar
indefinite medium properties within the interior of the conversion
structure.

[0063]The exemplary conversion structures 100 in FIGS. 1, 4, 7, and 10
provide an indefinite medium that is depicted as responding to an
evanescent electromagnetic wave to provide a non-evanescent
electromagnetic wave. In some embodiments (e.g. where the indefinite
medium is a reciprocal electromagnetic medium) the indefinite medium
alternately or additionally has a reciprocal response, i.e. the
indefinite medium responds to a non-evanescent electromagnetic wave to
provide an evanescent electromagnetic wave. In a reciprocal response of a
conversion structure 100 as in FIGS. 1, 4, 7, and 10, the non-evanescent
electromagnetic wave 130 and the propagating electromagnetic wave 125 may
be regarded as having spatially-reversed propagation directions (i.e.
propagating from right to left in the figures) and the evanescent
electromagnetic wave 120 may be regarded as having a spatially-reversed
exponential decay (i.e. having an exponential decay from right to left,
rather than from left to right as depicted). Thus, in a reciprocal
scenario, the conversion structure 100 is responsive to a
(leftwards-propagating) non-evanescent electromagnetic wave 130 at the
second surface region 112 to convey a propagating electromagnetic wave
125 from the second surface region to the first surface region 111 and to
provide an (leftwards-decaying) evanescent electromagnetic wave 120 at
the first surface region. As before, the evanescent electromagnetic wave
120 may be characterized by a first transverse wavevector kT.sup.(1)
as in equation (23) and the non-evanescent electromagnetic wave 130 may
be characterized by a second transverse wavevector kT.sup.(2) as in
equation (24). In the reciprocal scenario, when the indefinite medium is
a degenerate indefinite medium, the propagating electromagnetic wave 125
may propagate along a propagation direction that corresponds to a
direction antiparallel to the axial direction of the indefinite medium;
when the indefinite medium is a non-degenerate indefinite medium, the
propagating electromagnetic wave 125 may propagate along multiple
propagation directions--e.g. along at least two propagation directions,
each of the at least two directions having a substantially common angle
with respect to a direction antiparallel to the axial direction, or along
a plurality of propagation directions, the plurality of propagation
directions substantially composing a cone having a cone axis that
substantially coincides with a direction antiparallel to the axial
direction. In the depictions of FIGS. 1, 4, 7, and 10, the axial field
lines diverge geometrically as they proceed from the first surface region
111 to the second surface region 112; equivalently, the axial field lines
converge geometrically from the second surface 112 to the first surface
111. In the reciprocal scenario, this geometric convergence may
provide--for a propagating electromagnetic wave 125 characterized by a
transverse wavevector kT--a substantially continuous variation of
the transverse wavevector kT, from kT.sup.(2) at the second
surface region (as in equation (24), to match the transverse wavevector
of the non-evanescent electromagnetic wave 130) to kT.sup.(1) at the
first surface region (as in equation (23), to match the transverse
wavevector of the evanescent electromagnetic wave 120). Thus, the
geometric convergence of the axial field lines admits the conversion of a
non-evanescent electromagnetic wave 130 to an evanescent electromagnetic
wave 120, by supporting a propagating electromagnetic wave having a
substantially continuous variation of transverse wavevector from an
initial transverse wavevector that does not exceed a maximum wavevector
for non-evanescent waves to a final transverse wavevector that exceeds a
maximum wavevector for non-evanescent waves.

[0064]Some embodiments are responsive to an evanescent electromagnetic
wave to provide a non-evanescent electromagnetic wave (and/or vice versa,
in a reciprocal scenario) at a selected frequency/frequency band and/or a
selected polarization. The selected frequency or frequency band may be
selected from a range that includes radio frequencies, microwave
frequencies, millimeter- or submillimeter-wave frequencies, THz-wave
frequencies, optical frequencies (e.g. variously corresponding to soft
x-rays, extreme ultraviolet, ultraviolet, visible, near-infrared,
infrared, or far infrared light), etc. The selected polarization may be a
TE polarization, a TM polarization, a circular polarization, etc. (other
embodiments are responsive to an evanescent electromagnetic wave to
provide a non-evanescent electromagnetic wave-and/or vice versa, in a
reciprocal scenario--for any polarization, e.g. for unpolarized
electromagnetic energy).

[0065]Some embodiments are responsive to an evanescent electromagnetic
wave to a provide a non-evanescent electromagnetic wave (and/or vice
versa, in a reciprocal scenario) at a first frequency, and further
responsive to an evanescent electromagnetic wave to a provide a
non-evanescent electromagnetic wave (and/or vice versa, in a 25
reciprocal scenario) at a second frequency different than the first
frequency. For embodiments that recite first and second frequencies, the
first and second frequencies may be selected from the frequency
categories in the preceding paragraph. Moreover, for these embodiments,
the recitation of first and second frequencies may generally be replaced
by a recitation of first and second frequency bands, again selected from
30 the above frequency categories. These embodiments responsive at first
and second frequencies may include a indefinite medium having adjustable
electromagnetic properties. For example, the indefinite medium may have
electromagnetic properties that are adjustable (e.g. in response to an
external input or control signal) between first electromagnetic
properties and second electromagnetic properties, the first
electromagnetic properties providing an indefinite medium responsive to
an evanescent electromagnetic wave to provide a non-evanescent
electromagnetic wave (and/or vice versa) at the first frequency, and the
second electromagnetic properties providing an indefinite medium
responsive to an evanescent electromagnetic wave to provide a
non-evanescent electromagnetic wave (and/or vice versa) at the second
frequency. An indefinite medium with an adjustable electromagnetic
response may be implemented with variable metamaterials, e.g. as
described in R. A. Hyde et al, supra. Other embodiments responsive at
first and second frequencies may include an indefinite medium having a
frequency-dependent response to electromagnetic radiation, corresponding
to frequency-dependent constitutive parameters. For example, the
frequency-dependent response at a first frequency may be a response to an
evanescent electromagnetic wave to provide a non-evanescent
electromagnetic wave (and/or vice versa) at the first frequency, and the
frequency-dependent response at a second frequency may be a response to
an evanescent electromagnetic wave to provide a non-evanescent
electromagnetic wave (and/or vice versa) at the second frequency. An
indefinite medium having a frequency-dependent response to
electromagnetic radiation can be implemented with artificially-structured
materials such as metamaterials; for example, a first set of metamaterial
elements having a response at the first frequency may be interleaved with
a second set of metamaterial elements having a response at the second
frequency.

[0066]An illustrative embodiments is depicted as a process flow diagram in
FIG. 12. Flow 1200 includes operation 1210--coupling to an evanescent
electromagnetic wave at an input region. For example, a conversion
structure, such as that depicted as element 100 in FIG. 10, couples to an
evanescent electromagnetic wave 120 at a first surface region 111 of the
conversion structure. Flow 1200 includes operation 1220--responsive to
the coupling, propagating electromagnetic energy in an electromagnetic
field from a first surface within the input region to a second surface
within an output region, where the first and second surfaces are
substantially nonplanar and substantially non-concentric. For example,
the conversion structure 100 in FIG. 10 conveys a propagating
electromagnetic wave 125 from the first surface region 111 to the second
surface region 112. The first surface may be an exterior surface of the
input region, or at least partially within an interior portion of the
input region (e.g. corresponding to a conversion structure having an
input surface region 111 that is at least partially interior to the
conversion structure). The second surface may be an exterior surface of
the output region, or at least partially within an interior portion of
the output region (e.g. corresponding to a conversion structure having an
output surface region 112 that is at least partially interior to the
conversion structure). Operation 1220 includes sub-operation
1221--inducing a first polarization in a first direction, the first
polarization positively corresponding to a first component of the
electromagnetic field in the first direction--and sub-operation
1222--inducing a second polarization in a second direction perpendicular
to the first direction, the second polarization negatively corresponding
to a second component of the electromagnetic field in the second
direction. For example, a conversion structure 100 as in FIG. 10 may
provide an indefinite medium having an indefinite permittivity, and the
indefinite permittivity may correspond to an electric susceptibility that
is positive in a first direction and negative in a second direction
(where the first and second directions may correspond to axial and
transverse directions of the indefinite medium, or vice versa). Then the
propagating electromagnetic wave 125 may induce an electric polarization
in the first direction that positively corresponds (in accordance with
the positive electric susceptibility) to an electric field component of
the propagating electromagnetic wave in the first direction, and further
induce an electric polarization in the second direction that negatively
corresponds (in accordance with the negative electric susceptibility) to
an electric field component of the propagating electromagnetic wave in
the second direction. In an alternate example, a conversion structure 100
as in FIG. 10 may provide an indefinite medium having an indefinite
permeability, and the indefinite permeability may correspond to a
magnetic susceptibility that is positive in a first direction and
negative in a second direction (where the first and second directions may
correspond to axial and transverse directions of the indefinite medium,
or vice versa). Then the propagating electromagnetic wave 125 may induce
a magnetic polarization in the first direction that positively
corresponds (in accordance with the positive magnetic susceptibility) to
a magnetic field component of the propagating electromagnetic wave in the
first direction, and further induce a magnetic polarization in the second
direction that negatively corresponds (in accordance with the negative
magnetic susceptibility) to a magnetic field component of the propagating
electromagnetic wave in the second direction. Operation 1220 optionally
further includes sub-operation 1223--inducing a third polarization in a
third direction, the third polarization positively corresponding to a
third component of the electromagnetic field in the third direction--and
sub-operation 1224--inducing a fourth polarization in a fourth direction
perpendicular to the third direction, the fourth polarization negatively
corresponding to a fourth component of the electromagnetic field in the
fourth direction. For example, a conversion structure 100 as in FIG. 10
may provide an indefinite medium having both an indefinite permittivity
and an indefinite permeability, the indefinite permittivity corresponding
to a electric susceptibility that is positive in a first direction and
negative in a second direction (where the first and second directions may
correspond to axial and transverse directions of the indefinite medium,
or vice versa) and the indefinite permeability corresponding to a
magnetic susceptibility that is positive in a third direction and
negative in a fourth direction (where the third and fourth directions may
correspond to axial and transverse directions of the indefinite medium,
or vice versa). Then the propagating electromagnetic wave 125 may induce:
(1) an electric polarization in the first direction that positively
corresponds (in accordance with the positive electric susceptibility) to
an electric field component of the propagating electromagnetic wave in
the first direction, (2) an electric polarization in the second direction
that negatively corresponds (in accordance with the negative electric
susceptibility) to an electric field component of the propagating
electromagnetic wave in the second direction, (3) a magnetic polarization
in the third direction that positively corresponds (in accordance with
the positive magnetic susceptibility) to a magnetic field component of
the propagating electromagnetic wave in the second direction, and (4) a
magnetic polarization in the fourth direction that negatively corresponds
(in accordance with the negative magnetic susceptibility) to a magnetic
field component of the propagating electromagnetic wave in the fourth
direction. Flow 1200 includes operation 1230--providing the propagated
electromagnetic energy as a non-evanescent electromagnetic wave at the
output region. For example, the conversion structure 100 of FIG. 10
provides a non-evanescent electromagnetic wave 130 at the second surface
region 112; the non-evanescent electromagnetic wave may be a
freely-propagating electromagnetic wave, e.g. a wave that is emitted by
and freely radiates from the second electromagnetic surface, or a
confinedly-propagating electromagnetic wave, e.g. a wave that is
transmitted by the second surface region into a propagating guided wave
mode (as in a waveguide, transmission line, optical fiber, etc.).

[0067]An illustrative embodiments is depicted as a process flow diagram in
FIG. 13. Flow 1300 includes operation 1310--receiving a non-evanescent
electromagnetic wave at an input region. For example, a conversion
structure, such as that depicted as element 100 in FIG. 10, receives (in
a reciprocal scenario to that of FIG. 10, as described previously) a
non-evanescent electromagnetic wave 130 at the second surface region 112.
Flow 1300 includes operation 1320--responsive to the receiving,
propagating electromagnetic energy in an electromagnetic field from a
first surface within the input region to a second surface within an
output region, where the first and second surfaces are substantially
nonplanar and substantially non-concentric. For example, the conversion
structure 100 in FIG. 10 conveys (in a reciprocal scenario to that of
FIG. 10, as described previously) a propagating electromagnetic wave 125
from the second surface region 112 to the first surface region 111. The
first surface may be an exterior surface of the input region, or at least
partially within an interior portion of the input region (e.g.
corresponding to a conversion structure having a second surface region
112 that is at least partially interior to the conversion structure). The
second surface may be an exterior surface of the output region, or at
least partially within an interior portion of the output region (e.g.
corresponding to a conversion structure having an first surface region
111 that is at least partially interior to the conversion structure).
Operation 1320 includes sub-operation 1321--inducing a first polarization
in a first direction, the first polarization positively corresponding to
a first component of the electromagnetic field in the first
direction--and sub-operation 1322--inducing a second polarization in a
second direction perpendicular to the first direction, the second
polarization negatively corresponding to a second component of the
electromagnetic field in the second direction. For example, a conversion
structure 100 as in FIG. 10 may provide an indefinite medium having an
indefinite permittivity, and the indefinite permittivity may correspond
to an electric susceptibility that is positive in a first direction and
negative in a second direction (where the first and second directions may
correspond to axial and transverse directions of the indefinite medium,
or vice versa). Then the propagating electromagnetic wave 125 may induce
an electric polarization in the first direction that positively
corresponds (in accordance with the positive electric susceptibility) to
an electric field component of the propagating electromagnetic wave in
the first direction, and further induce an electric polarization in the
second direction that negatively corresponds (in accordance with the
negative electric susceptibility) to an electric field component of the
propagating electromagnetic wave in the second direction. In an alternate
example, a conversion structure 100 as in FIG. 10 may provide an
indefinite medium having an indefinite permeability, and the indefinite
permeability may correspond to a magnetic susceptibility that is positive
in a first direction and negative in a second direction (where the first
and second directions may correspond to axial and transverse directions
of the indefinite medium, or vice versa). Then the propagating
electromagnetic wave 125 may induce a magnetic polarization in the first
direction that positively corresponds (in accordance with the positive
magnetic susceptibility) to a magnetic field component of the propagating
electromagnetic wave in the first direction, and further induce a
magnetic polarization in the second direction that negatively corresponds
(in accordance with the negative magnetic susceptibility) to a magnetic
field component of the propagating electromagnetic wave in the second
direction. Operation 1320 optionally further includes sub-operation
1323--inducing a third polarization in a third direction, the third
polarization positively corresponding to a third component of the
electromagnetic field in the third direction--and sub-operation
1324--inducing a fourth polarization in a fourth direction perpendicular
to the third direction, the fourth polarization negatively corresponding
to a fourth component of the electromagnetic field in the fourth
direction. For example, a conversion structure 100 as in FIG. 10 may
provide an indefinite medium having both an indefinite permittivity and
an indefinite permeability, the indefinite permittivity corresponding to
a electric susceptibility that is positive in a first direction and
negative in a second direction (where the first and second directions may
correspond to axial and transverse directions of the indefinite medium,
or vice versa) and the indefinite permeability corresponding to a
magnetic susceptibility that is positive in a third direction and
negative in a fourth direction (where the third and fourth directions may
correspond to axial and transverse directions of the indefinite medium,
or vice versa). Then the propagating electromagnetic wave 125 may induce:
(1) an electric polarization in the first direction that positively
corresponds (in accordance with the positive electric susceptibility) to
an electric field component of the propagating electromagnetic wave in
the first direction, (2) an electric polarization in the second direction
that negatively corresponds (in accordance with the negative electric
susceptibility) to an electric field component of the propagating
electromagnetic wave in the second direction, (3) a magnetic polarization
in the third direction that positively corresponds (in accordance with
the positive magnetic susceptibility) to a magnetic field component of
the propagating electromagnetic wave in the second direction, and (4) a
magnetic polarization in the fourth direction that negatively corresponds
(in accordance with the negative magnetic susceptibility) to a magnetic
field component of the propagating electromagnetic wave in the fourth
direction. Flow 1300 includes operation 1330--coupling the propagated
electromagnetic energy to an evanescent electromagnetic wave at the
output region. For example, the conversion structure 100 of FIG. 10
provides (in a reciprocal scenario to that of FIG. 10, as described
previously) an evanescent electromagnetic wave 120 at the first surface
region 111 (the evanescent wave having an exponential decay away from the
conversion structure for this reciprocal scenario, not decaying towards
the conversion structure as depicted).

[0068]With reference now to FIG. 14, an illustrative embodiment is
depicted as a system block diagram. The system 1400 includes an
evanescent conversion unit 1420 optionally coupled to a control unit
1440. The evanescent conversion unit 1420 may include a conversion
structure such as that depicted as element 100 in FIGS. 1-11. The
conversion structure may be a variable conversion structure, such as a
variable metamaterial responsive to one or more control inputs to vary
one or more operating characteristics (operating frequency, operating
wave polarization, effective coordinate transformation for a
transformation medium, etc), and the control unit 1440 may include
control circuitry that provides one or more control inputs to the
variable conversion structure. The evanescent conversion unit 1420 may
further include a positioning structure (e.g. with one or more piezo
stages, nanopositioners, conveyors/turntables, or other actuators) having
one or more control inputs to vary a position/orientation of the
conversion structure and/or vary a position/orientation of a sample or
target in relation to the conversion structure (e.g. within an evanescent
range of the conversion structure), and the control unit 1440 may include
control circuitry that provides the one or more control inputs to the
positioning structure, optionally in response to a feedback signal from
the positioning structure (e.g. a cantilever force feedback). The
evanescent conversion unit 1420 may include one or more optical
components, e.g. positioned to deliver electromagnetic energy to an input
surface of the conversion structure, receive electromagnetic energy from
an output surface of the conversion structure, deliver electromagnetic
energy to a sample or target positioned within an evanescent range of the
conversion structure, and/or receive electromagnetic energy from a sample
or target positioned within an evanescent range of the conversion
structure; and the control unit 1440 may include control circuitry that
provides one or more control inputs to the one or more optical components
(e.g. to control orientations, focusing characteristics, aperture sizes,
etc.). The system optionally includes an input unit 1410 coupled to the
evanescent conversion unit 1420 (e.g. to deliver electromagnetic energy
to the evanescent conversion unit 1420); the input unit may include an
electromagnetic source (e.g. an antenna, laser, or transducer) as well as
input circuitry and/or optical components such as modulators, phase
adjusters, etc. The system optionally includes an output unit 1430
coupled to the evanescent conversion unit 1420 (e.g. to receive
electromagnetic energy from the evanescent conversion unit 1420); the
output unit may include an electromagnetic detector (e.g. a CCD array,
photomultiplier, etc.) as well as output circuitry and/or optical
components such as demodulators, phase adjusters, spectral analyzers,
image processing circuitry, etc.

[0069]All of the above U.S. patents, U.S. patent application publications,
U.S. patent applications, foreign patents, foreign patent applications
and non-patent publications referred to in this specification and/or
listed in any Application Data Sheet, are incorporated herein by
reference, to the extent not inconsistent herewith.

[0070]One skilled in the art will recognize that the herein described
components (e.g., steps), devices, and objects and the discussion
accompanying them are used as examples for the sake of conceptual clarity
and that various configuration modifications are within the skill of
those in the art. Consequently, as used herein, the specific exemplars
set forth and the accompanying discussion are intended to be
representative of their more general classes. In general, use of any
specific exemplar herein is also intended to be representative of its
class, and the non-inclusion of such specific components (e.g., steps),
devices, and objects herein should not be taken as indicating that
limitation is desired.

[0071]With respect to the use of substantially any plural and/or singular
terms herein, those having skill in the art can translate from the plural
to the singular and/or from the singular to the plural as is appropriate
to the context and/or application. The various singular/plural
permutations are not expressly set forth herein for sake of clarity.

[0072]While particular aspects of the present subject matter described
herein have been shown and described, it will be apparent to those
skilled in the art that, based upon the teachings herein, changes and
modifications may be made without departing from the subject matter
described herein and its broader aspects and, therefore, the appended
claims are to encompass within their scope all such changes and
modifications as are within the true spirit and scope of the subject
matter described herein. Furthermore, it is to be understood that the
invention is defined by the appended claims. It will be understood by
those within the art that, in general, terms used herein, and especially
in the appended claims (e.g., bodies of the appended claims) are
generally intended as "open" terms (e.g., the term "including" should be
interpreted as "including but not limited to," the term "having" should
be interpreted as "having at least," the term "includes" should be
interpreted as "includes but is not limited to," etc.). It will be
further understood by those within the art that if a specific number of
an introduced claim recitation is intended, such an intent will be
explicitly recited in the claim, and in the absence of such recitation no
such intent is present. For example, as an aid to understanding, the
following appended claims may contain usage of the introductory phrases
"at least one" and "one or more" to introduce claim recitations. However,
the use of such phrases should not be construed to imply that the
introduction of a claim recitation by the indefinite articles "a" or "an"
limits any particular claim containing such introduced claim recitation
to inventions containing only one such recitation, even when the same
claim includes the introductory phrases "one or more" or "at least one"
and indefinite articles such as "a" or "an" (e.g., "a" and/or "an" should
typically be interpreted to mean "at least one" or "one or more"); the
same holds true for the use of definite articles used to introduce claim
recitations. In addition, even if a specific number of an introduced
claim recitation is explicitly recited, those skilled in the art will
recognize that such recitation should typically be interpreted to mean at
least the recited number (e.g., the bare recitation of "two recitations,"
without other modifiers, typically means at least two recitations, or two
or more recitations). Furthermore, in those instances where a convention
analogous to "at least one of A, B, and C, etc." is used, in general such
a construction is intended in the sense one having skill in the art would
understand the convention (e.g., "a system having at least one of A, B,
and C" would include but not be limited to systems that have A alone, B
alone, C alone, A and B together, A and C together, B and C together,
and/or A, B, and C together, etc.). In those instances where a convention
analogous to "at least one of A, B, or C, etc." is used, in general such
a construction is intended in the sense one having skill in the art would
understand the convention (e.g., "a system having at least one of A, B,
or C" would include but not be limited to systems that have A alone, B
alone, C alone, A and B together, A and C together, B and C together,
and/or A, B, and C together, etc.). It will be further understood by
those within the art that virtually any disjunctive word and/or phrase
presenting two or more alternative terms, whether in the description,
claims, or drawings, should be understood to contemplate the
possibilities of including one of the terms, either of the terms, or both
terms. For example, the phrase "A or B" will be understood to include the
possibilities of "A" or "B" or "A and B."

[0073]With respect to the appended claims, those skilled in the art will
appreciate that recited operations therein may generally be performed in
any order. Examples of such alternate orderings may include overlapping,
interleaved, interrupted, reordered, incremental, preparatory,
supplemental, simultaneous, reverse, or other variant orderings, unless
context dictates otherwise. With respect to context, even terms like
"responsive to," "related to," or other past-tense adjectives are
generally not intended to exclude such variants, unless context dictates
otherwise.

[0074]While various aspects and embodiments have been disclosed herein,
other aspects and embodiments will be apparent to those skilled in the
art. The various aspects and embodiments disclosed herein are for
purposes of illustration and are not intended to be limiting, with the
true scope and spirit being indicated by the following claims.